#363541

order	512
nilpotency class	3
isoclinism	isoclinic class of #189517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#423 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

976 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:

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

Polycyclic presentation:

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

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, 270737);