#6358089

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:

(1,2,8,3)(4,14,15,16)(5,9,11,12)(6,10,7,13)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,32)(29,42)(30,43)(31,37)(33,39)(34,40)(35,41)(38,45)(44,48)(46,47);(1,3,7,13,8,2,6,10)(4,5,12,14,15,11,9,16)(17,19,22,32,23,29,21,31)(18,24,26,28,27,37,36,42)(20,30,34,45,33,44,35,46)(25,38,41,43,39,47,40,48);(1,4,7,12,8,15,6,9)(2,5,10,14,3,11,13,16)(19,29)(20,33)(24,37)(25,39)(28,42)(31,32)(34,35)(40,41);(1,5)(2,9)(3,12)(4,10)(6,16)(7,14)(8,11)(13,15)(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);(1,6,8,7)(2,10,3,13)(4,9,15,12)(5,14,11,16)(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=i,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,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, 6265285);