#6815748

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814811
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1691 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18)(19,28)(20,25)(21,37)(22,26)(23,27)(24,31)(29,41)(30,42)(32,38)(33,35)(34,47)(36,40)(39,45)(43,48)(44,46);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,18)(19,28)(20,33)(21,26)(22,37)(23,27)(24,32)(25,35)(29,41)(30,44)(31,38)(34,40)(36,47)(39,43)(42,46)(45,48);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(17,19)(18,24,27,38)(20,30)(21,31)(22,32)(23,29)(25,39,40,44)(26,28,37,41)(33,42,47,48)(34,45)(35,46)(36,43);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(19,29)(24,38)(28,41)(30,43)(31,32)(39,44)(42,48)(45,46);(3,13)(7,10)(11,12)(15,16)(17,20,21,34,23,36,22,35)(18,25,37,47,27,40,26,33)(19,30,31,45,29,43,32,46)(24,39,41,48,38,44,28,42)

Polycyclic presentation:

2^9|c^2=fgh,d^2=f,e^2=hi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di

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, 6722944);