#111228

order	512
nilpotency class	3
isoclinism	isoclinic class of #111226
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

880 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=i,e^2=h,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=eh,g^b=ghi,d^c=di,e^c=eh

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, 18424);