#6930789

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
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	#25797	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1695 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6837985);