#6676019

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1819 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6583215);