#198467

order	512
nilpotency class	3
isoclinism	isoclinic class of #198162
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2052 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105663);