#6679531

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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	#36018	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1649 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

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, 6586727);