#6654613

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632068
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	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1361 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=f,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,g^c=gi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6561809);