#219638

order	512
nilpotency class	3
isoclinism	isoclinic class of #219638
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#535
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

488 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ef,d^2=e,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=dhi,f^b=fh,d^c=di,g^c=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, 126834);