#6556565

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1385 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7)(4,8)(5,6)(17,18,22,26)(19,27,29,44)(20,31,35,47)(21,36,33,25)(23,37,39,30)(24,40,42,34)(28,41,32,38)(43,48,46,45);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,30,22,29,33,37)(18,23,25,27,26,39,36,44)(20,32,34,48,35,28,40,45)(24,41,31,46,42,38,47,43);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,33,40,22,35,21,34)(18,24,36,47,26,42,25,31)(19,28,37,48,29,32,30,45)(23,38,44,46,39,41,27,43);(1,4,5,3)(2,8,6,7)(9,11)(10,14)(12,15)(13,16)(19,29)(23,39)(27,44)(28,32)(30,37)(38,41)(43,46)(45,48);(11,15)(14,16)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fi,d^2=fgi,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^d=ehi

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, 6463761);