#393646

order	512
nilpotency class	3
isoclinism	isoclinic class of #32944
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄³

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1276 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)(3,6)(4,5)(7,8)(9,10)(11,16,23,33)(12,25)(13,17)(14,28)(15,19)(18,26)(20,29)(21,35,36,31)(22,30,37,40)(24,39,38,32)(27,34);(9,11)(10,16,19,33)(12,21)(13,22)(14,24)(15,23)(17,30,34,40)(18,31,25,35)(20,32,28,39)(26,36)(27,37)(29,38);(2,5)(6,8)(9,12,14,13,15,26,29,27)(10,17,20,25,19,34,28,18)(11,21,24,22,23,36,38,37)(16,30,32,35,33,40,39,31);(1,3,4,7)(2,6,5,8)(9,13,14,26,15,27,29,12)(10,18,20,17,19,25,28,34)(11,22,24,36,23,37,38,21)(16,31,32,30,33,35,39,40)

Polycyclic presentation:

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

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, 300842);