#6596295

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3524	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2976 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=i,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,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, 6503491);