#219630

order	512
nilpotency class	3
isoclinism	isoclinic class of #219624
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	C₂×#127	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

542 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ef,d^2=ehi,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,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, 126826);