#110235

order	512
nilpotency class	3
isoclinism	isoclinic class of #9443
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

650 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efi,d^2=hi,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,d^b=dh,e^b=ei,f^b=fi,d^c=di,e^c=eh,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17431);