#6514506

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500450
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	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1817 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)(13,14,20,26)(15,22,29,35)(16,21,28,36)(17,23,31,38)(18,25,19,24)(27,37,41,44)(30,34,42,43)(32,40,33,39);(5,6)(7,10)(8,9)(11,12)(13,15,19,28,20,29,18,16)(14,21,25,35,26,36,24,22)(17,27,33,42,31,41,32,30)(23,34,40,44,38,43,39,37);(2,4)(6,9)(7,11)(13,16,18,29,20,28,19,15)(14,22,24,36,26,35,25,21)(17,30,32,41,31,42,33,27)(23,37,39,43,38,44,40,34);(5,7,8,11)(6,10,9,12)(13,17,20,31)(14,23,26,38)(15,27,29,41)(16,30,28,42)(18,32,19,33)(21,34,36,43)(22,37,35,44)(24,39,25,40);(6,9)(10,12)(17,31)(23,38)(27,41)(30,42)(32,33)(34,43)(37,44)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6421702);