#6667602

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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

2728 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6574798);