#201840

order	512
nilpotency class	3
isoclinism	isoclinic class of #201687
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1182 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,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, 109036);