#6685447

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	#24397	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1875 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)(7,10)(8,9)(11,12)(13,14,19,23)(15,20,18,32)(16,26,28,40)(17,22,31,37)(21,24,35,29)(25,33,30,36)(27,39,42,43)(34,38,44,41);(1,2)(3,4)(6,9)(7,11)(13,15,19,18)(14,20,23,32)(16,24,28,29)(17,30,31,25)(21,26,35,40)(22,36,37,33)(27,41,42,38)(34,43,44,39);(2,4)(13,16,18,29,19,28,15,24)(14,21,20,26,23,35,32,40)(17,27,30,41,31,42,25,38)(22,34,33,39,37,44,36,43);(6,9)(10,12)(13,17,19,31)(14,22,23,37)(15,25,18,30)(16,27,28,42)(20,33,32,36)(21,34,35,44)(24,38,29,41)(26,39,40,43);(2,4)(5,7,8,11)(6,10,9,12)(16,28)(17,31)(21,35)(22,37)(24,29)(25,30)(26,40)(33,36)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=i,e^2=f,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,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, 6592643);