#114637

order	512
nilpotency class	3
isoclinism	isoclinic class of #114631
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

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

1142 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1142 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)(11,20,24,35)(12,17)(13,18)(14,29)(15,19)(16,22,31,23)(21,37,39,32)(25,40,38,30)(26,33)(27,36)(28,34);(2,6)(3,4)(9,11)(10,16)(12,25)(13,23)(14,22)(15,24)(17,32)(18,35)(19,31)(20,29)(21,27)(26,39)(28,38)(30,33)(34,37)(36,40);(2,7)(3,4)(6,8)(9,12,15,28)(10,17,19,34)(11,21,24,39)(13,27,14,26)(16,30,31,40)(18,36,29,33)(20,37,35,32)(22,25,23,38)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=h,f^2=i,g^2=h,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, 21833);