#6668105

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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 ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2730 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,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, 6575301);