#213803

order	512
nilpotency class	3
isoclinism	isoclinic class of #213798
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2431	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

872 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cei,f^b=fh,g^b=gh,d^c=di,f^c=fh,g^c=gh,g^d=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120999);