#6806355

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682500
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1252 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:

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

Polycyclic presentation:

2^9|b^2=fhi,c^2=fg,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei

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, 6713551);