#72300531

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 10421 conjugacy classes of subgroups; only the 3425 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

(5,6)(7,10)(8,12)(9,11)(13,14)(15,18)(16,19)(17,20)(21,22)(23,26)(24,27)(25,28);(1,2)(3,4)(5,7,9,12)(6,8,11,10)(13,15,17,19)(14,18,20,16)(21,23,25,27)(22,26,28,24);(1,2,3,4)(5,7,9,12)(6,10,11,8)(13,15)(14,16)(17,19)(18,20)(21,24,25,26)(22,27,28,23);(1,2)(3,4)(6,11)(7,12)(13,15)(14,18)(16,20)(17,19)(21,23)(22,24)(25,27)(26,28);(5,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28);(5,7)(6,10)(8,11)(9,12)(13,16)(14,19)(15,20)(17,18)(21,24)(22,27)(23,28)(25,26)

Polycyclic presentation:

2^10|b^2=gh,c^2=ghij,b^a=bg,c^a=cgi,c^b=cij,d^a=dhi,d^b=dgh,d^c=dj,e^a=egh,e^b=egh,f^b=fgh,f^c=fgh,f^d=fhi,f^e=fgh

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300575
maximal normal subgroup of	#72300575

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.7*F.8);;
gap> SetPower(coll, 3, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.7*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);