#6802710

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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

1529 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)(17,18,21,26,22,25,34,39)(19,23,29,37,28,36,44,47)(20,30,31,38,32,40,33,24)(27,41,43,46,42,48,45,35);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14)(17,19)(18,23)(20,27)(21,29)(22,28)(24,35)(25,36)(26,37)(30,41)(31,43)(32,42)(33,45)(34,44)(38,46)(39,47)(40,48);(3,8)(5,7)(10,13)(14,15)(17,20,22,32)(18,24,25,38)(19,27,28,42)(21,33,34,31)(23,35,36,46)(26,40,39,30)(29,45,44,43)(37,48,47,41);(2,6)(5,7)(19,28)(23,36)(27,42)(29,44)(35,46)(37,47)(41,48)(43,45);(9,11,12,16)(10,14,13,15)(18,25)(23,36)(24,38)(26,39)(30,40)(35,46)(37,47)(41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6709906);