#201319

order	512
nilpotency class	3
isoclinism	isoclinic class of #201094
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1087 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=i,e^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,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, 108515);