#114645

order	512
nilpotency class	3
isoclinism	isoclinic class of #114631
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1116 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1116 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) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,d^c=di,e^c=eh,f^c=fi,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, 21841);