#6936950

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885129
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	#37553	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1583 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)(3,6)(4,5)(7,8)(17,18,22,26,23,27,21,35)(19,24,31,39,29,38,30,45)(20,32,33,41,34,40,36,25)(28,42,44,47,43,48,46,37);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,29)(18,24,27,38)(20,28,34,43)(21,30,22,31)(25,37,41,47)(26,39,35,45)(32,42,40,48)(33,44,36,46);(1,3,4,7)(2,6,5,8)(10,13)(11,15)(17,20,23,34)(18,25,27,41)(19,28,29,43)(21,33,22,36)(24,37,38,47)(26,40,35,32)(30,44,31,46)(39,48,45,42);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(19,29)(24,38)(28,43)(30,31)(37,47)(39,45)(42,48)(44,46);(10,13)(14,16)(17,21,23,22)(18,26,27,35)(19,30,29,31)(20,33,34,36)(24,39,38,45)(25,40,41,32)(28,44,43,46)(37,48,47,42)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fi,d^2=fh,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6844146);