#6686747

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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1891 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,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, 6593943);