#6560722

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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

1370 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6467918);