#6569346

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1373 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=gi,c^2=f,d^2=h,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ei,d^c=di,e^d=ehi

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, 6476542);