#6908552

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1703 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,e^d=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, 6815748);