#6560142

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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	#38092	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1629 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)(17,18)(19,23)(20,32)(21,25)(22,27)(24,35)(26,34)(28,44)(29,38)(30,39)(31,40)(33,47)(36,42)(37,46)(41,45)(43,48);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,33)(21,30)(22,31)(24,41)(25,39)(26,38)(27,40)(28,36)(29,34)(32,47)(35,45)(37,43)(42,44)(46,48);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,20,34,48,22,36,21,35)(18,24,25,42,27,43,26,32)(19,28,29,45,31,33,30,46)(23,37,39,47,40,41,38,44);(9,11)(10,14)(12,15)(13,16)(17,21,22,34)(18,25,27,26)(19,29,31,30)(20,35,36,48)(23,38,40,39)(24,42,43,32)(28,45,33,46)(37,44,41,47);(11,15)(14,16)(17,21,22,34)(18,25,27,26)(19,30,31,29)(20,35,36,48)(23,39,40,38)(24,42,43,32)(28,46,33,45)(37,47,41,44)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6467338);