#6685475

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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	#29736	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1877 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)(15,20,18,33)(16,26,28,43)(17,22,32,39)(21,24,36,29)(25,34,30,37)(27,42,31,44)(35,40,38,41);(1,2)(3,4)(6,9)(7,11)(13,15,19,18)(14,20,23,33)(16,24,28,29)(17,30,32,25)(21,26,36,43)(22,37,39,34)(27,41,31,40)(35,44,38,42);(2,4)(13,16,18,29,19,28,15,24)(14,21,20,26,23,36,33,43)(17,31,30,40,32,27,25,41)(22,38,34,44,39,35,37,42);(6,9)(10,12)(13,17)(14,22)(15,25)(16,27)(18,30)(19,32)(20,34)(21,35)(23,39)(24,40)(26,42)(28,31)(29,41)(33,37)(36,38)(43,44);(2,4)(5,7,8,11)(6,10,9,12)(16,28)(17,32)(21,36)(22,39)(24,29)(25,30)(26,43)(34,37)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,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,d^c=di,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, 6592671);