#357532

order	512
nilpotency class	3
isoclinism	isoclinic class of #23444
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

4478 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^b=hi,d^c=di

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, 264728);