#6716702

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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1313 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)(4,8)(5,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,29,28)(27,33,36,31)(30,32,38,37)(34,39,40,35);(1,3,5,4)(2,6,7,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27)(26,31)(28,33)(29,36)(30,39)(32,34)(35,38)(37,40);(2,7)(6,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,26,29,28)(27,34,36,40)(30,37,38,32)(31,35,33,39);(10,16)(14,17)(15,22)(21,23)(25,28,29,26)(27,33,36,31)(30,37,38,32)(34,39,40,35);(1,4,5,3)(2,8,7,6)(11,19)(14,17)(18,24)(21,23)(27,35)(30,38)(31,40)(32,37)(33,34)(36,39)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=f,d^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,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, 6623898);