#6676691

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	#42880	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1885 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)(7,10)(8,12)(9,11)(13,14,18,16)(15,23,27,20)(17,29,31,21)(19,22,33,30)(24,28,35,26)(25,34,41,37)(32,43,44,36)(38,40,39,42);(5,7,10,6)(8,12,9,11)(13,15)(14,20)(16,23)(17,25)(18,27)(19,26)(21,34)(22,35)(24,30)(28,33)(29,37)(31,41)(32,40)(36,39)(38,43)(42,44);(2,4)(13,16,18,14)(15,24,27,35)(17,21,31,29)(19,22,33,30)(20,26,23,28)(25,39,41,38)(32,43,44,36)(34,42,37,40);(5,8,10,9)(6,11,7,12)(13,17,18,31)(14,21,16,29)(15,25,27,41)(19,32,33,44)(20,34,23,37)(22,36,30,43)(24,38,35,39)(26,40,28,42);(5,9,10,8)(6,11,7,12)(15,26)(19,33)(20,35)(22,30)(23,24)(25,40)(27,28)(32,44)(34,39)(36,43)(37,38)(41,42)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6583887);