#201794

order	512
nilpotency class	3
isoclinism	isoclinic class of #201701
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1250 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=i,d^2=hi,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,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, 108990);