#205728

order	512
nilpotency class	3
isoclinism	isoclinic class of #205569
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1212 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)(3,4)(5,8)(6,7)(9,10,14,19,15,20,13,18)(11,21,24,16,25,31,23,32)(12,26,29,33,27,34,28,17)(22,36,39,35,37,30,38,40);(1,3,4,2,6,8,5,7)(9,11)(10,16)(12,22)(13,24)(14,23)(15,25)(17,30)(18,31)(19,21)(20,32)(26,35)(27,37)(28,39)(29,38)(33,36)(34,40);(9,12,13,28,15,27,14,29)(10,17,19,26,20,33,18,34)(11,22,24,39,25,37,23,38)(16,30,21,35,32,36,31,40);(1,4,6,5)(2,3,7,8)(12,27)(17,33)(22,37)(26,34)(28,29)(30,36)(35,40)(38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112924);