#6815749

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

1689 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)(19,29)(20,32)(21,27)(22,26)(23,30)(24,28)(25,34)(31,45)(33,47)(35,40)(36,48)(37,42)(38,46)(39,44)(41,43);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,19)(18,20)(21,36)(22,32)(23,25)(24,40)(26,29)(27,43)(28,34)(30,35)(31,38)(33,46)(37,41)(39,47)(42,48)(44,45);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(17,20,22,29)(18,25)(19,30)(21,33,37,45)(23,34,24,35)(26,40)(27,41)(28,32)(31,46)(36,42)(38,43,39,48)(44,47);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(18,26)(20,29)(27,42)(28,30)(33,45)(34,35)(43,48)(44,46);(3,13)(7,10)(11,12)(15,16)(17,21,24,39,22,37,23,38)(18,27,28,44,26,42,30,46)(19,31,25,41,32,47,40,36)(20,33,35,48,29,45,34,43)

Polycyclic presentation:

2^9|c^2=fgh,d^2=fi,e^2=h,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, 6722945);