#6908002

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

2024 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,30)(21,26)(22,36)(23,27)(25,32)(28,39)(29,37)(31,43)(33,44)(34,40)(35,41)(38,45)(42,48)(46,47);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19,23,22)(18,24,27,36)(20,28,33,32)(21,29,35,34)(25,30,39,44)(26,37,41,40)(31,42,46,45)(38,43,48,47);(1,2)(3,5)(4,6)(7,8)(11,16)(13,15)(17,20,22,32,23,33,19,28)(18,25,24,30,27,39,36,44)(21,31,34,45,35,46,29,42)(26,38,37,43,41,48,40,47);(2,6)(5,7)(10,14)(13,15)(17,21)(18,26)(19,29)(20,31)(22,34)(23,35)(24,37)(25,38)(27,41)(28,42)(30,43)(32,45)(33,46)(36,40)(39,48)(44,47);(3,8)(5,7)(17,19,23,22)(18,24,27,36)(20,28,33,32)(21,34,35,29)(25,30,39,44)(26,40,41,37)(31,45,46,42)(38,47,48,43)

Polycyclic presentation:

2^9|b^2=i,c^2=g,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^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, 6815198);