#361109

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₄⋊C₄
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

784 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,g^c=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, 268305);