#6798297

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37680	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1770 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)(4,8)(5,7)(10,13)(11,15)(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);(1,3,6,2)(4,8,5,7)(9,10,12,13)(11,14,15,16);(10,13)(14,16)(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);(1,4,6,5)(2,7,3,8)(9,11,12,15)(10,14,13,16)(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);(1,5,6,4)(2,7,3,8)(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|b^2=f,c^2=g,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,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, 6705493);