#6520120

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500120
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1760 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)(13,14,17,22,18,21,31,37)(15,23,26,33,25,38,30,19)(16,20,29,36,28,35,43,44)(24,39,40,34,27,41,42,32);(5,6)(7,10)(8,9)(11,12)(13,15,18,25)(14,19,21,33)(16,27,28,24)(17,30,31,26)(20,34,35,32)(22,38,37,23)(29,40,43,42)(36,39,44,41);(1,2,3,4)(6,9)(7,11)(13,16,18,28)(14,20,21,35)(15,24,25,27)(17,29,31,43)(19,32,33,34)(22,36,37,44)(23,39,38,41)(26,40,30,42);(5,7,8,11)(6,10,9,12)(14,21)(16,28)(19,33)(22,37)(23,38)(24,27)(29,43)(40,42);(6,9)(10,12)(15,25)(16,28)(19,33)(20,35)(23,38)(26,30)(29,43)(36,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^c=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, 6427316);