#201897

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

1096 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,10,6,9,4,8)(3,7,11,14,13,16,12,15)(17,18)(19,23,29,38)(20,30,32,24)(21,33)(22,26)(25,35)(27,36)(28,40,34,37)(31,39,46,44)(41,48)(42,43)(45,47);(1,3,4,11,6,13,5,12)(2,7,10,15,9,16,8,14)(17,19,22,29)(18,23)(20,27,32,42)(21,34,35,28)(24,36)(25,40)(26,38)(30,43)(31,45,46,41)(33,37)(39,48)(44,47);(1,4,6,5)(2,8,9,10)(3,11,13,12)(7,14,16,15)(17,20,22,32)(18,24,26,30)(19,27,29,42)(21,31,35,46)(23,36,38,43)(25,39,33,44)(28,41,34,45)(37,47,40,48);(1,4,6,5)(2,8,9,10)(3,12,13,11)(7,15,16,14)(17,21,22,35)(18,25,26,33)(19,28,29,34)(20,31,32,46)(23,37,38,40)(24,39,30,44)(27,41,42,45)(36,47,43,48)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=h,g^2=hi,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, 109093);