#7054072

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2137 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,19,24)(15,20,18,23)(16,27,30,34)(17,22,33,39)(21,28,37,25)(26,35,32,40)(29,43,31,44)(36,42,38,41);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15,19,18)(14,20,24,23)(16,28,30,25)(17,26,33,32)(21,27,37,34)(22,35,39,40)(29,42,31,41)(36,43,38,44);(5,7,8,12)(6,9,10,11)(13,16)(14,21)(15,25)(17,31)(18,28)(19,30)(20,34)(22,38)(23,27)(24,37)(26,42)(29,33)(32,41)(35,43)(36,39)(40,44);(2,4)(7,12)(9,11)(13,17)(14,22)(15,26)(16,29)(18,32)(19,33)(20,35)(21,36)(23,40)(24,39)(25,41)(27,43)(28,42)(30,31)(34,44)(37,38);(6,10)(9,11)(13,15,19,18)(14,23,24,20)(16,25,30,28)(17,32,33,26)(21,27,37,34)(22,35,39,40)(29,42,31,41)(36,44,38,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6961268);