#453876

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

544 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)(3,6)(4,5)(7,8)(9,10,16,29,14,13,15,19)(11,17,22,32,23,21,30,33)(12,18,28,40,26,25,27,35)(20,31,24,34,37,36,39,38);(9,11,16,22,14,23,15,30)(10,17,19,33,13,21,29,32)(12,24,28,37,26,39,27,20)(18,34,35,31,25,38,40,36);(2,5)(6,8)(9,12,15,27,14,26,16,28)(10,18,19,35,13,25,29,40)(11,20,22,24,23,37,30,39)(17,31,32,34,21,36,33,38);(1,3,4,7)(2,6,5,8)(9,13)(10,14)(11,21)(12,25)(15,29)(16,19)(17,23)(18,26)(20,36)(22,33)(24,38)(27,40)(28,35)(30,32)(31,37)(34,39)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361072);