#6561246

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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ 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)(17,18,22,27)(19,28,30,39)(20,32,35,41)(21,25,33,26)(23,36,38,31)(24,40,42,34)(29,44,45,37)(43,47,48,46);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,31,22,30,33,36)(18,23,26,28,27,38,25,39)(20,29,34,46,35,45,40,47)(24,37,32,43,42,44,41,48);(10,13)(14,16)(17,20,33,40,22,35,21,34)(18,24,25,41,27,42,26,32)(19,29,36,47,30,45,31,46)(23,37,39,48,38,44,28,43);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(19,30)(23,38)(28,39)(29,45)(31,36)(37,44)(43,48)(46,47);(11,15)(14,16)(17,21,22,33)(18,25,27,26)(19,31,30,36)(20,34,35,40)(23,39,38,28)(24,41,42,32)(29,46,45,47)(37,48,44,43)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=gi,d^2=fgi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=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, 6468442);