#6786355

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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	#38032	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1535 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)(19,23)(20,31)(21,25)(22,27)(24,28)(26,30)(29,38)(32,44)(33,40)(34,37)(35,41)(36,42)(39,43)(45,48)(46,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,23,27,26)(20,28,34,33)(21,35,36,29)(24,37,40,31)(25,41,42,38)(32,45,46,43)(39,44,48,47);(2,6)(3,8)(10,13)(14,16)(17,20,30,33,22,34,19,28)(18,24,23,37,27,40,26,31)(21,32,35,45,36,46,29,43)(25,39,38,47,42,48,41,44);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21,22,36)(18,25,27,42)(19,29,30,35)(20,32,34,46)(23,38,26,41)(24,39,40,48)(28,43,33,45)(31,44,37,47);(10,13)(14,16)(17,19,22,30)(18,23,27,26)(20,33,34,28)(21,35,36,29)(24,31,40,37)(25,41,42,38)(32,43,46,45)(39,47,48,44)

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=fhi,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,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, 6693551);