#72327240

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 9347 conjugacy classes of subgroups; only the 3397 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:

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

Polycyclic presentation:

2^10|a^2=j,c^2=j,e^2=j,b^a=bg,c^b=cgh,d^a=dhi,d^c=dhi,e^b=ehi,e^d=ehi,f^a=fj,f^b=fh,f^d=fh,f^e=fgi

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72327309
maximal normal subgroup of	#72327309

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, 1, F.10);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.7*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);