#6356258

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356241
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1524 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fh,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^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6263454);