#6816022

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2157 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^d=ei

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, 6723218);