#113133

order	512
nilpotency class	3
isoclinism	isoclinic class of #113119
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

819 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,d^c=di,f^c=fh,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20329);