#7101532

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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

1315 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)(17,18,24,37,23,28,22,27)(19,29,21,25,32,39,31,40)(20,26,34,41,36,43,35,42)(30,44,33,38,46,48,45,47);(9,10)(11,15)(12,14)(13,16)(17,19,23,32)(18,25,28,40)(20,30,36,46)(21,24,31,22)(26,38,43,47)(27,39,37,29)(33,34,45,35)(41,44,42,48);(1,3,5,4)(2,6,8,7)(9,11,12,16)(10,13,14,15)(17,20,23,36)(18,26,28,43)(19,30,32,46)(21,33,31,45)(22,35,24,34)(25,38,40,47)(27,42,37,41)(29,44,39,48);(1,3,5,4)(2,7,8,6)(11,16)(13,15)(17,21,22,19,23,31,24,32)(18,25,27,29,28,40,37,39)(20,33,35,30,36,45,34,46)(26,38,42,44,43,47,41,48);(1,4,5,3)(2,6,8,7)(10,14)(13,15)(17,22,23,24)(18,27,28,37)(19,21,32,31)(20,34,36,35)(25,39,40,29)(26,41,43,42)(30,45,46,33)(38,44,47,48)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=ghi,d^2=fg,e^2=gi,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,e^d=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, 7008728);