#393871

order	512
nilpotency class	3
isoclinism	isoclinic class of #32941
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#390
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1176 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=efh,c^2=f,d^2=g,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gi,e^b=ei,h^b=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301067);