#219596

order	512
nilpotency class	3
isoclinism	isoclinic class of #219590
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

505 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ef,d^2=ei,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=dh,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, 126792);