#6629270

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628896
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,e^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,e^b=ei,h^c=hi,g^d=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536466);