#232698

order	512
nilpotency class	3
isoclinism	isoclinic class of #232688
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	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1346 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 1346 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) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=hi,e^2=h,f^2=i,g^2=h,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=gh,e^c=ei,f^c=fi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139894);