#6786393

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682460
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	#38035	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1532 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)(3,7)(4,6)(5,8)(17,18,22,27)(19,23,29,36)(20,31,33,41)(21,25,32,26)(24,39,40,34)(28,42,45,47)(30,37,43,38)(35,46,48,44);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,35)(25,37)(26,38)(27,36)(31,42)(32,43)(33,45)(34,44)(39,46)(40,48)(41,47);(2,6)(3,8)(10,13)(14,16)(17,20,32,39,22,33,21,34)(18,24,25,41,27,40,26,31)(19,28,43,46,29,45,30,44)(23,35,37,47,36,48,38,42);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(19,29)(20,33)(23,36)(24,40)(30,43)(31,41)(34,39)(37,38);(10,13)(14,16)(17,21,22,32)(18,25,27,26)(19,30,29,43)(20,34,33,39)(23,37,36,38)(24,41,40,31)(28,44,45,46)(35,47,48,42)

Polycyclic presentation:

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

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, 6693589);