#214053

order	512
nilpotency class	3
isoclinism	isoclinic class of #214040
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	#2619	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

798 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:

(1,2,5,11,6,10,4,9)(3,7,12,15,13,8,14,16)(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);(1,3)(2,7)(4,14)(5,12)(6,13)(8,10)(9,16)(11,15)(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)(2,8)(4,12)(5,14)(6,13)(7,10)(9,16)(11,15)(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)(2,9,10,11)(3,12,13,14)(7,15,8,16)(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,d^b=di,f^b=fh,g^b=gh,f^c=fh,g^c=ghi,g^d=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121249);