#6485704

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

1427 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)(17,18,23,28)(19,29,33,37)(20,26,36,42)(21,27,22,25)(24,31,39,32)(30,43,46,47)(34,40,35,41)(38,45,48,44);(2,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19,23,33)(18,24,28,39)(20,30,36,46)(21,31,22,32)(25,37,27,29)(26,38,42,48)(34,44,35,45)(40,47,41,43);(10,14)(15,16)(18,25)(21,22)(24,37)(26,40)(27,28)(29,39)(31,32)(34,35)(38,47)(41,42)(43,48)(44,45);(9,11,13,12)(10,15,14,16)(17,20)(18,26)(19,30)(21,35)(22,34)(23,36)(24,38)(25,40)(27,41)(28,42)(29,43)(31,45)(32,44)(33,46)(37,47)(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)(19,31,33,32)(20,34,36,35)(24,29,39,37)(26,40,42,41)(30,44,46,45)(38,47,48,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=h,e^2=fgi,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,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, 6392900);