#6837258

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	C₂×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1384 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

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, 6744454);