#6358053

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358021
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

1800 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)(5,10,9,13)(6,11,7,12)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,31)(29,42)(30,38)(32,43)(33,39)(34,40)(35,41)(37,45)(44,48)(46,47);(1,3,6,11,8,2,7,12)(4,5,13,14,15,9,10,16)(17,19,22,31,23,32,21,30)(18,24,26,28,27,38,36,43)(20,29,34,45,33,44,35,46)(25,37,41,42,39,47,40,48);(1,4,7,10,8,15,6,13)(2,9,11,14,3,5,12,16)(20,33)(25,39)(29,44)(34,35)(37,47)(40,41)(42,48)(45,46);(1,5)(2,10)(3,13)(4,12)(6,14)(7,16)(8,9)(11,15)(17,20)(18,25)(19,29)(21,35)(22,34)(23,33)(24,37)(26,41)(27,39)(28,42)(30,46)(31,45)(32,44)(36,40)(38,47)(43,48);(1,6,8,7)(2,11,3,12)(4,13,15,10)(5,16,9,14)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,34,33,35)(24,28,38,43)(25,40,39,41)(29,45,44,46)(37,48,47,42)

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,e^b=ei,d^c=dhi,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, 6265249);