#6624800

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1407 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)(9,10)(11,15)(12,16)(13,14)(17,18,23,28)(19,29,33,39)(20,34,35,42)(21,27,22,26)(24,32,40,31)(25,36,41,37)(30,38,44,48)(43,45,47,46);(2,6)(7,8)(10,14)(11,12)(17,19,22,32,23,33,21,31)(18,24,27,39,28,40,26,29)(20,30,37,46,35,44,36,45)(25,38,34,47,41,48,42,43);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,20,22,37,23,35,21,36)(18,25,27,34,28,41,26,42)(19,30,32,46,33,44,31,45)(24,38,39,47,40,48,29,43);(1,4,5,3)(2,8,6,7)(11,12)(15,16);(9,12,13,11)(10,16,14,15)(20,35)(25,41)(30,44)(34,42)(36,37)(38,48)(43,47)(45,46)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fi,d^2=fgh,e^2=h,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^c=ei,e^d=eh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6531996);