#359440

order	512
nilpotency class	3
isoclinism	isoclinic class of #184831
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

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

3824 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)(9,10,13,18,14,17,26,31)(11,15,21,29,20,28,36,39)(12,22,23,32,24,16,25,30)(19,33,35,40,34,27,37,38);(2,4)(5,6)(7,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,27)(17,28)(18,29)(22,33)(23,35)(24,34)(25,37)(26,36)(30,38)(31,39)(32,40);(6,8)(9,12,14,24)(10,16,17,22)(11,19,20,34)(13,25,26,23)(15,27,28,33)(18,32,31,30)(21,37,36,35)(29,40,39,38);(2,4)(11,20)(15,28)(19,34)(21,36)(27,33)(29,39)(35,37)(38,40)

Polycyclic presentation:

2^9|a^2=h,c^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266636);