#232889

order	512
nilpotency class	3
isoclinism	isoclinic class of #232889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1352 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1352 conjugacy classes of subgroups — too many to draw. Only its 190 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) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=dh,e^b=eh,g^b=ghi,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, 140085);