#356860

order	512
nilpotency class	3
isoclinism	isoclinic class of #17411
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1105 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)(3,6)(4,5)(7,8)(9,10,15,19,14,18,30,13)(11,20,24,29,23,16,31,22)(12,25,28,26,27,17,33,34)(21,32,38,36,37,35,40,39);(9,11)(10,16,18,20)(12,21)(13,29,19,22)(14,23)(15,24)(17,32,25,35)(26,39,34,36)(27,37)(28,38)(30,31)(33,40);(2,5)(6,8)(9,12,15,28,14,27,30,33)(10,17,19,34,18,25,13,26)(11,21,24,38,23,37,31,40)(16,32,22,36,20,35,29,39);(1,3,4,7)(2,6,5,8)(9,13,30,18,14,19,15,10)(11,22,31,16,23,29,24,20)(12,26,33,25,27,34,28,17)(21,36,40,32,37,39,38,35)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=fg,g^2=i,b^a=be,c^a=cf,h^a=hi,d^b=de,e^b=ei,h^d=hi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264056);