#6556540

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1391 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)(4,8)(5,6)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,32)(25,36)(30,38)(31,40)(33,39)(35,42)(37,41)(43,46)(44,48)(45,47);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,33,23,31,21,32)(18,24,26,40,28,39,27,29)(20,30,36,46,37,44,35,45)(25,38,34,47,42,48,41,43);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,22,36,23,37,21,35)(18,25,26,34,28,42,27,41)(19,30,33,46,31,44,32,45)(24,38,40,47,39,48,29,43);(1,4,5,3)(2,8,6,7)(9,11)(10,14)(12,15)(13,16)(19,31)(24,39)(29,40)(30,44)(32,33)(38,48)(43,47)(45,46);(11,15)(14,16)(17,21,23,22)(18,26,28,27)(19,32,31,33)(20,35,37,36)(24,40,39,29)(25,34,42,41)(30,45,44,46)(38,47,48,43)

Polycyclic presentation:

2^9|b^2=g,c^2=fi,d^2=fg,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, 6463736);