#6964795

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885065
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1521 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=fi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6871991);