#216023

order	512
nilpotency class	3
isoclinism	isoclinic class of #163726
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

780 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,25,27,22)(20,23)(21,24)(26,31,32,29)(28,30)(33,34)(35,42)(36,40)(37,39)(38,43)(41,45)(44,46)(47,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(33,35,36,43)(34,38,40,42)(37,44,45,48)(39,47,41,46);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(34,39)(35,43)(37,45)(38,46)(40,41)(42,47);(2,8)(6,9)(7,16)(14,15)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(34,40)(35,43)(39,41)(44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=eh,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=di,f^c=fi,g^c=ghi

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, 123219);