#6676641

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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	#42788	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1884 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)(14,19)(16,27)(20,33)(22,31)(23,36)(24,28)(29,41)(40,43);(5,7,10,6)(8,12,9,11)(13,14)(15,23)(16,21)(17,19)(18,31)(20,26)(22,30)(24,39)(25,36)(27,34)(28,42)(29,44)(32,33)(35,40)(37,43)(38,41);(2,4)(13,15,17,25)(14,20,19,33)(16,28,27,24)(18,32,30,26)(21,37,34,35)(22,36,31,23)(29,40,41,43)(38,39,44,42);(5,8,10,9)(6,11,7,12)(13,16,17,27)(14,21,19,34)(15,24,25,28)(18,29,30,41)(20,35,33,37)(22,38,31,44)(23,39,36,42)(26,40,32,43);(5,9,10,8)(6,11,7,12)(14,22)(18,30)(19,31)(20,36)(21,38)(23,33)(26,32)(29,41)(34,44)(35,42)(37,39)(40,43)

Polycyclic presentation:

2^9|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=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, 6583837);