#6485771

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1390 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^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, 6392967);