#214013

order	512
nilpotency class	3
isoclinism	isoclinic class of #214000
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 ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

861 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:

(1,2,5,10,6,9,4,8)(3,11,12,16,13,7,14,15)(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);(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)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(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);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(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=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,f^b=fh,g^b=gh,f^c=fh,g^c=ghi,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, 121209);