#363542

order	512
nilpotency class	3
isoclinism	isoclinic class of #189674
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

954 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:

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

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,d^b=di,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, 270738);