#363540

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×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#423 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

998 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)(11,22)(12,26,30,40)(13,17)(14,31)(15,20)(16,21)(18,29,35,28)(19,25)(23,37,39,32)(24,33)(27,36,38,34);(1,2)(3,6)(4,5)(7,8)(9,11,14,13,16,25,15,24)(10,17,20,22,21,33,31,19)(12,23,28,27,30,39,29,38)(18,32,26,36,35,37,40,34);(2,5)(6,8)(9,12,14,28,16,30,15,29)(10,18,20,26,21,35,31,40)(11,23,13,27,25,39,24,38)(17,32,22,36,33,37,19,34);(1,3,4,7)(2,6,5,8)(9,13,14,25,16,24,15,11)(10,19,20,17,21,22,31,33)(12,27,28,39,30,38,29,23)(18,34,26,32,35,36,40,37)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=eg,e^2=i,f^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, 270736);