#6763447

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23965	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1642 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)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18)(19,26)(20,31)(21,35)(22,27)(23,30)(24,28)(25,37)(29,38)(32,41)(33,46)(34,39)(36,42)(40,45)(43,48)(44,47);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,23,27,26)(20,32,34,28)(21,29,37,36)(24,39,41,31)(25,38,35,42)(33,45,47,43)(40,46,48,44);(2,6)(3,7)(10,13)(14,16)(17,20,19,28,22,34,30,32)(18,24,23,31,27,41,26,39)(21,33,29,43,37,47,36,45)(25,40,38,44,35,48,42,46);(9,11,12,15)(10,14,13,16)(17,21)(18,25)(19,29)(20,33)(22,37)(23,38)(24,40)(26,42)(27,35)(28,43)(30,36)(31,44)(32,45)(34,47)(39,46)(41,48);(3,7)(5,8)(17,19,22,30)(18,26,27,23)(20,28,34,32)(21,36,37,29)(24,39,41,31)(25,38,35,42)(33,45,47,43)(40,44,48,46)

Polycyclic presentation:

2^9|b^2=i,c^2=fgh,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,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, 6670643);