#6358091

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358057
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

1792 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)(3,13)(4,15)(5,10)(6,11)(7,12)(8,14)(9,16)(17,18,23,27)(19,28,29,42)(20,25,33,39)(21,26,22,36)(24,31,37,32)(30,43,44,48)(34,40,35,41)(38,46,47,45);(1,3,6,9,7,15,5,14)(2,8,10,4,12,16,11,13)(17,19,21,31,23,29,22,32)(18,24,36,42,27,37,26,28)(20,30,35,46,33,44,34,45)(25,38,40,48,39,47,41,43);(1,4,6,16,7,13,5,8)(2,9,10,15,12,14,11,3)(19,29)(20,33)(24,37)(25,39)(28,42)(31,32)(34,35)(40,41);(1,2)(3,8)(4,9)(5,11)(6,10)(7,12)(13,14)(15,16)(17,20)(18,25)(19,30)(21,35)(22,34)(23,33)(24,38)(26,41)(27,39)(28,43)(29,44)(31,46)(32,45)(36,40)(37,47)(42,48);(1,5,7,6)(2,10,12,11)(3,9,15,14)(4,8,13,16)(17,21,23,22)(18,26,27,36)(19,31,29,32)(20,34,33,35)(24,28,37,42)(25,40,39,41)(30,45,44,46)(38,48,47,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=fh,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,e^b=ei,d^c=dhi,e^d=eh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265287);