#6786330

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682532
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

1509 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)(3,7)(4,6)(5,8)(17,18,21,25)(19,27,29,23)(20,31,32,47)(22,35,34,26)(24,38,39,33)(28,42,43,46)(30,37,45,41)(36,44,40,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,36)(25,27)(26,37)(31,46)(32,43)(33,44)(34,45)(35,41)(38,48)(39,40)(42,47);(2,6)(3,8)(10,13)(14,16)(17,20,22,33,21,32,34,38)(18,24,26,31,25,39,35,47)(19,28,30,44,29,43,45,48)(23,36,37,46,27,40,41,42);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(18,25)(19,29)(24,39)(26,35)(28,43)(30,45)(31,47)(44,48);(10,13)(14,16)(19,29)(20,32)(23,27)(24,39)(30,45)(31,47)(33,38)(37,41)

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,d^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,e^c=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, 6693526);