#232911

order	512
nilpotency class	3
isoclinism	isoclinic class of #232905
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1258 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 1258 conjugacy classes of subgroups — too many to draw. Only its 198 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=dh,e^b=eh,g^b=ghi,d^c=dh,e^c=ei,f^c=fi,g^c=ghi

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, 140107);