#6567319

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1317 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)(3,7)(4,6)(5,8)(9,10,14,13)(11,16,12,15)(17,18,22,26)(19,27,31,44)(20,32,34,47)(21,36,28,25)(23,37,39,30)(24,40,42,35)(29,41,33,38)(43,48,46,45);(1,3,5,4)(2,6,8,7)(9,11,14,12)(10,15,13,16)(17,19,28,37,22,31,21,30)(18,23,36,44,26,39,25,27)(20,33,40,45,34,29,35,48)(24,41,47,43,42,38,32,46);(1,3,5,4)(2,7,8,6)(9,10,14,13)(11,16,12,15)(17,20,21,35,22,34,28,40)(18,24,25,32,26,42,36,47)(19,29,30,45,31,33,37,48)(23,38,27,43,39,41,44,46);(1,3,5,4)(2,7,8,6)(9,12,14,11)(10,15,13,16)(20,34)(24,42)(29,33)(32,47)(35,40)(38,41)(43,46)(45,48);(1,4,5,3)(2,6,8,7)(9,13,14,10)(11,16,12,15)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=f,d^2=fgi,e^2=fgi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^c=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, 6474515);