#6709679

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667484
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1249 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)(3,6)(4,5)(7,8)(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23);(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,26,30,35)(27,32,37,33)(28,38,39,40)(29,31,36,34);(2,5)(6,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,31)(28,39)(29,36)(32,38)(33,40)(34,35);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(25,27,30,37)(26,32,35,33)(28,36,39,29)(31,38,34,40);(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28,30,39)(26,33,35,32)(27,36,37,29)(31,38,34,40)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=fgi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6616875);