#6501541

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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

1572 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)(17,18,21,26)(19,28,30,23)(20,25,32,41)(22,35,34,27)(24,36,40,33)(29,43,45,38)(31,39,47,44)(37,46,42,48);(2,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,19,21,30)(18,23,26,28)(20,29,32,45)(22,31,34,47)(24,37,40,42)(25,38,41,43)(27,39,35,44)(33,46,36,48);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,20)(18,24)(19,29)(21,32)(22,36)(23,37)(25,35)(26,40)(27,41)(28,42)(30,45)(31,48)(33,34)(38,44)(39,43)(46,47);(1,4,5,3)(2,8,6,7)(11,16)(13,15)(17,20)(18,25)(19,29)(21,32)(22,33)(23,38)(24,35)(26,41)(27,40)(28,43)(30,45)(31,46)(34,36)(37,44)(39,42)(47,48);(10,14)(13,15)(20,32)(24,40)(25,41)(29,45)(33,36)(37,42)(38,43)(46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6408737);