#201563

order	512
nilpotency class	3
isoclinism	isoclinic class of #201404
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	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1183 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:

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

Polycyclic presentation:

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

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, 108759);