#198429

order	512
nilpotency class	3
isoclinism	isoclinic class of #198135
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1187 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=h,d^2=i,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=ehi,f^c=fhi,g^c=ghi

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, 105625);