#201998

order	512
nilpotency class	3
isoclinism	isoclinic class of #201918
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	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2002 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)(3,7,8,5)(4,6)(9,10,14,18,15,19,13,29)(11,20,24,31,22,35,23,16)(12,25,27,17,28,32,26,33)(21,30,38,34,36,40,37,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16,19,31)(12,21)(13,23)(14,24)(15,22)(17,30,33,40)(18,20,29,35)(25,39,32,34)(26,37)(27,38)(28,36);(1,3)(2,5)(4,8)(6,7)(9,12,15,28)(10,17,19,33)(11,21,22,36)(13,26,14,27)(16,30,31,40)(18,32,29,25)(20,34,35,39)(23,37,24,38);(11,22)(16,31)(20,35)(21,36)(23,24)(30,40)(34,39)(37,38)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,f^b=fi,e^c=ei,f^c=fh,g^c=gi

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, 109194);