#6485751

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1429 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,29)(20,26)(21,27)(22,25)(23,28)(24,31)(30,43)(32,38)(33,42)(34,40)(35,41)(36,39)(37,46)(44,47)(45,48);(2,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19,23,32)(18,24,28,39)(20,30,33,44)(21,31,22,36)(25,29,27,38)(26,37,42,48)(34,45,35,46)(40,43,41,47);(10,14)(15,16)(18,25)(20,33)(21,22)(24,29)(26,41)(27,28)(30,44)(31,36)(37,47)(38,39)(40,42)(43,48);(9,11,13,12)(10,15,14,16)(17,20)(18,26)(19,30)(21,35)(22,34)(23,33)(24,37)(25,40)(27,41)(28,42)(29,43)(31,46)(32,44)(36,45)(38,47)(39,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,14,16)(17,21,23,22)(18,27,28,25)(19,31,32,36)(20,34,33,35)(24,38,39,29)(26,40,42,41)(30,45,44,46)(37,43,48,47)

Polycyclic presentation:

2^9|b^2=i,d^2=h,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,d^c=di,e^c=ei,f^c=fi,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, 6392947);