#201097

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

1056 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)(3,10,11,14)(4,7,12,13)(5,8)(6,9)(15,16)(17,18)(19,28)(20,33,36,25)(21,26)(22,37)(23,27)(24,31)(29,42,46,48)(30,39)(32,43)(34,41,35,40)(38,45,47,44);(1,3,5,13,6,11,15,7)(2,4,8,14,9,12,16,10)(17,19,22,31,23,32,21,30)(18,24,26,28,27,39,37,43)(20,29,34,44,36,46,35,45)(25,38,41,42,33,47,40,48);(1,4)(2,7)(3,8)(5,14)(6,12)(9,13)(10,15)(11,16)(17,20,23,36)(18,25,27,33)(19,29,32,46)(21,35,22,34)(24,38,39,47)(26,41,37,40)(28,42,43,48)(30,45,31,44);(17,21,23,22)(18,26,27,37)(19,30,32,31)(20,34,36,35)(24,28,39,43)(25,40,33,41)(29,44,46,45)(38,48,47,42)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=h,e^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, 108293);