#111441

order	512
nilpotency class	3
isoclinism	isoclinic class of #111436
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	#60639	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1132 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1132 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,f^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,e^b=ei,f^b=fi,g^b=ghi,f^c=fi,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, 18637);