#6764475

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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	#2615	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1496 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)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,27,28,41)(20,31,32,25)(22,35,34,26)(24,37,36,30)(29,43,42,40)(33,39,47,46)(38,44,45,48);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(18,23)(19,28)(20,32)(26,35)(30,37)(33,47)(38,45)(40,43);(10,14)(11,15)(17,19,22,30,21,28,34,37)(18,24,26,27,23,36,35,41)(20,29,33,44,32,42,47,48)(25,38,39,40,31,45,46,43);(2,5)(6,8)(17,20,21,32)(18,25,23,31)(19,29,28,42)(22,33,34,47)(24,38,36,45)(26,39,35,46)(27,40,41,43)(30,44,37,48);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(19,28)(20,32)(24,36)(25,31)(27,41)(30,37)(33,47)(39,46)

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=i,e^2=fi,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,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, 6671671);