#6540872

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1308 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)(17,18,23,27)(19,28,32,42)(20,33,34,47)(21,26,22,37)(24,30,39,31)(25,35,40,36)(29,43,44,38)(41,45,48,46);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,30,23,32,22,31)(18,24,37,42,27,39,26,28)(20,29,35,45,34,44,36,46)(25,38,47,48,40,43,33,41);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,20,22,36,23,34,21,35)(18,25,26,33,27,40,37,47)(19,29,31,46,32,44,30,45)(24,38,28,41,39,43,42,48);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(20,34)(25,40)(29,44)(33,47)(35,36)(38,43)(41,48)(45,46);(1,4,5,3)(2,6,8,7)(11,15)(14,16)(17,21,23,22)(18,26,27,37)(19,30,32,31)(20,35,34,36)(24,28,39,42)(25,33,40,47)(29,45,44,46)(38,41,43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,d^2=fgi,e^2=fg,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=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, 6448068);