#6908228

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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1728 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,e^c=ei,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, 6815424);