#6805578

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1440 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)(17,18,22,28)(19,27,23,24)(20,31,34,38)(21,35,37,26)(25,29,41,32)(30,39,36,42)(33,46,47,44)(40,45,48,43);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14)(17,19,22,23)(18,24,28,27)(20,32,34,29)(21,30,37,36)(25,31,41,38)(26,39,35,42)(33,45,47,43)(40,44,48,46);(3,8)(5,7)(10,13)(14,15)(17,20,23,32,22,34,19,29)(18,25,27,31,28,41,24,38)(21,33,36,45,37,47,30,43)(26,40,42,44,35,48,39,46);(2,6)(5,7)(17,21,22,37)(18,26,28,35)(19,30,23,36)(20,33,34,47)(24,39,27,42)(25,40,41,48)(29,43,32,45)(31,44,38,46);(9,11,12,16)(10,14,13,15)(17,19,22,23)(18,27,28,24)(20,32,34,29)(21,36,37,30)(25,38,41,31)(26,39,35,42)(33,43,47,45)(40,44,48,46)

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=g,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,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, 6712774);