#201698

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, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1254 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)(19,23)(20,30,32,24)(21,33)(22,26)(25,34)(27,35,40,43)(28,41)(29,37)(31,38,45,44)(36,42)(39,47,48,46);(1,3)(2,6,8,13)(4,9)(5,10)(7,12,14,16)(11,15)(17,19,22,29)(18,23,26,37)(20,27,32,40)(21,28,34,42)(24,35,30,43)(25,36,33,41)(31,39,45,48)(38,46,44,47);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,12,13,16)(17,20)(18,24)(19,27)(21,31)(22,32)(23,35)(25,38)(26,30)(28,39)(29,40)(33,44)(34,45)(36,46)(37,43)(41,47)(42,48);(17,21,22,34)(18,25,26,33)(19,28,29,42)(20,31,32,45)(23,36,37,41)(24,38,30,44)(27,39,40,48)(35,46,43,47)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=h,d^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, 108894);