#111930

order	512
nilpotency class	3
isoclinism	isoclinic class of #111915
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

856 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 856 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,g^b=ghi,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, 19126);