#6930781

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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1675 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)(17,18,22,26)(19,27,28,42)(20,24,32,40)(21,34,36,25)(23,35,37,30)(29,43,31,45)(33,48,47,41)(38,46,39,44);(1,2,4,5)(3,6,7,8)(9,11,12,16)(10,13,14,15)(19,28)(23,37)(27,42)(29,31)(30,35)(38,39)(43,45)(44,46);(3,7)(6,8)(11,16)(13,15)(17,19,22,28)(18,23,26,37)(20,31,32,29)(21,35,36,30)(24,39,40,38)(25,42,34,27)(33,44,47,46)(41,43,48,45);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20,22,32)(18,24,26,40)(19,29,28,31)(21,33,36,47)(23,38,37,39)(25,41,34,48)(27,43,42,45)(30,44,35,46);(2,5)(6,8)(20,32)(24,40)(29,31)(33,47)(38,39)(41,48)(43,45)(44,46)

Polycyclic presentation:

2^9|a^2=i,b^2=f,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, 6837977);