#219584

order	512
nilpotency class	3
isoclinism	isoclinic class of #219576
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

507 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=efi,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=dh,f^b=fh,g^c=gh

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, 126780);