#7187471

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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1402 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=fg,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,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, 7094667);