#201113

order	512
nilpotency class	3
isoclinism	isoclinic class of #201112
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1211 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)(19,27)(20,31,33,24)(21,34)(22,26)(23,30)(25,36)(28,41,43,48)(29,42)(32,39,35,40)(37,44,47,46)(38,45);(1,3)(2,6,8,13)(4,9)(5,10)(7,12,14,16)(11,15)(17,19,21,30,22,29,36,45)(18,23,25,27,26,38,34,42)(20,28,35,46,33,43,32,44)(24,37,40,41,31,47,39,48);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,12,13,16)(17,20,22,33)(18,24,26,31)(19,28,29,43)(21,35,36,32)(23,37,38,47)(25,40,34,39)(27,41,42,48)(30,46,45,44);(2,8)(3,10)(7,14)(9,15)(19,29)(23,38)(27,42)(28,43)(30,45)(37,47)(41,48)(44,46)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,e^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,g^c=ghi

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, 108309);