#6990451

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885065
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1643 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)(9,10,12,14)(11,15,16,13)(17,18)(19,27)(20,31)(21,25)(22,35)(23,29)(24,33)(26,34)(28,43)(30,38)(32,40)(36,39)(37,47)(41,45)(42,46)(44,48);(1,3)(2,5)(4,7)(6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,37)(25,27)(26,38)(31,48)(32,44)(33,45)(34,46)(35,42)(36,47)(39,41)(40,43);(1,3)(2,4)(5,7)(6,8)(11,16)(13,15)(17,20)(18,24)(19,28)(21,32)(22,36)(23,37)(25,39)(26,40)(27,41)(29,44)(30,47)(31,35)(33,34)(38,43)(42,48)(45,46);(1,4,6,5)(2,3,7,8)(10,14)(13,15)(18,25)(19,29)(20,32)(26,35)(30,46)(33,36)(37,41)(43,48);(1,5,6,4)(2,3,7,8)(19,29)(23,27)(28,44)(30,46)(37,41)(38,42)(43,48)(45,47)

Polycyclic presentation:

2^9|a^2=fi,b^2=h,d^2=fhi,e^2=fhi,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, 6897647);