#6698907

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1777 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

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, 6606103);