#127220

order	512
nilpotency class	4
isoclinism	isoclinic class of #127215
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#342
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

617 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=gh,f^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,h^a=hi,c^b=cg,e^b=eh,d^c=di,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34416);