#7101485

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1317 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=ghi,d^2=fgi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei

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, 7008681);