#6357725

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357716
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1837 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=di,e^c=eh,e^d=eh

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, 6264921);