#6554556

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1759 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=fi,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6461752);