#369409

order	512
nilpotency class	3
isoclinism	isoclinic class of #196350
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1430 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)(3,7)(4,6)(5,8)(9,10,14,18,13,17,20,28)(11,19,23,35,22,15,27,30)(12,16,25,32,24,31,26,33)(21,34,37,40,36,29,38,39);(1,3,5,4)(2,6,8,7)(9,11,20,27,13,22,14,23)(10,15,28,35,17,19,18,30)(12,21,26,38,24,36,25,37)(16,29,33,40,31,34,32,39);(1,3,5,4)(2,7,8,6)(9,12,13,24)(10,16)(11,21)(14,26,20,25)(15,29,19,34)(17,31)(18,33)(22,36)(23,38)(27,37)(28,32)(30,40,35,39);(1,4,5,3)(2,7,8,6)(10,17)(15,19)(16,31)(18,28)(29,34)(30,35)(32,33)(39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=efg,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,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, 276605);