#6653882

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632088
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

1460 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)(11,15)(14,16)(17,18,21,26)(19,28,29,23)(20,32,33,25)(22,24,35,27)(30,37,42,41)(31,36,43,38)(34,40,45,39)(44,47,48,46);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,29)(18,23,26,28)(20,30,33,42)(22,31,35,43)(24,38,27,36)(25,37,32,41)(34,44,45,48)(39,47,40,46);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(18,24)(19,29)(22,35)(23,36)(25,39)(26,27)(28,38)(30,42)(32,40)(34,45)(37,46)(41,47);(2,5)(6,8)(10,13)(14,16)(17,20,21,33)(18,25,26,32)(19,30,29,42)(22,34,35,45)(23,37,28,41)(24,39,27,40)(31,44,43,48)(36,46,38,47);(9,11,12,15)(10,14,13,16)(19,29)(20,33)(23,28)(25,32)(31,43)(34,45)(36,38)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6561078);