#119330

order	512
nilpotency class	3
isoclinism	isoclinic class of #119328
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

552 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 126 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 — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,e^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=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, 26526);