#6621115

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1733 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6528311);