#201566

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	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1216 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,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, 108762);