#72300525

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 9369 conjugacy classes of subgroups; only the 3289 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|b^2=g,d^2=h,e^2=g,b^a=bi,c^b=cg,d^b=dh,d^c=dghj,e^c=eg,e^d=egh,f^a=fh,f^c=fgij,f^d=fh

Conjugacy classes & character table

124 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);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);