#201254

order	512
nilpotency class	3
isoclinism	isoclinic class of #201112
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	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1090 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:

(1,2,5,8,6,9,4,13)(3,7,12,14,11,15,10,16)(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);(1,3,5,10,6,11,4,12)(2,7,13,14,9,15,8,16)(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);(17,20)(18,24)(19,28)(21,35)(22,33)(23,37)(25,40)(26,31)(27,41)(29,43)(30,46)(32,36)(34,39)(38,47)(42,48)(44,45);(1,4,6,5)(2,8,9,13)(3,10,11,12)(7,14,15,16)(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,d^2=i,e^2=hi,g^2=i,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, 108450);