#6989662

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907977
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

1489 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,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=dfhi,e^c=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6896858);