#6485654

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

Polycyclic presentation:

2^9|c^2=i,d^2=hi,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,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, 6392850);