#6992899

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885065
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1601 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,21,25)(19,27,29,23)(20,31,33,42)(22,36,35,26)(24,37,41,34)(28,44,32,48)(30,39,46,43)(38,45,40,47);(1,3,7,4)(2,5,8,6)(9,11,12,16)(10,13,14,15)(17,19,21,29)(18,23,25,27)(20,32,33,28)(22,30,35,46)(24,40,41,38)(26,39,36,43)(31,44,42,48)(34,47,37,45);(1,4,7,3)(2,5,8,6)(11,16)(13,15)(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);(1,5,7,6)(2,3,8,4)(10,14)(13,15)(18,25)(19,29)(24,41)(26,36)(28,32)(30,46)(31,42)(45,47);(1,6,7,5)(2,3,8,4)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fhi,d^2=fh,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, 6900095);