#6993261

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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1549 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)(3,7)(4,6)(5,8)(17,18,22,26,21,25,35,42)(19,27,30,44,29,23,37,39)(20,31,34,41,33,43,36,24)(28,45,46,38,32,48,47,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,23)(20,32)(21,29)(22,30)(24,40)(25,27)(26,39)(28,33)(31,45)(34,47)(35,37)(36,46)(38,41)(42,44)(43,48);(3,8)(5,7)(9,11,14,10)(12,15,13,16)(17,20,21,33)(18,24,25,41)(19,28,29,32)(22,36,35,34)(23,38,27,40)(26,43,42,31)(30,47,37,46)(39,45,44,48);(2,6)(5,7)(9,12,14,13)(10,15,11,16)(18,25)(19,29)(24,41)(26,42)(28,32)(30,37)(31,43)(46,47);(9,13,14,12)(10,15,11,16)

Polycyclic presentation:

2^9|a^2=g,c^2=fhi,d^2=fh,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,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, 6900457);