#220214

order	512
nilpotency class	3
isoclinism	isoclinic class of #220210
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#466	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

832 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127410);