#6683647

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23854	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2280 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,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, 6590843);