#232663

order	512
nilpotency class	3
isoclinism	isoclinic class of #232654
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	C₂³×C₄	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1413 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 1413 conjugacy classes of subgroups — too many to draw. Only its 206 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)(3,11,14,15)(4,7,5,9)(8,12,16,13)(17,18)(19,22)(20,23)(21,24)(25,29)(26,28)(27,30)(31,32);(2,7)(3,12)(4,5)(9,10)(11,15)(13,14)(20,27)(22,28)(23,30)(25,31);(1,3,6,14)(2,8,10,16)(4,12,5,13)(7,11,9,15)(17,19)(18,22)(20,25)(21,26)(23,29)(24,28)(27,31)(30,32);(1,4,6,5)(2,9,10,7)(3,12,14,13)(8,15,16,11)(17,20,21,27)(18,23,24,30)(19,25,26,31)(22,29,28,32)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=hi,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,g^a=ghi,c^b=cg,d^b=dhi,e^b=eh,f^b=fh,g^b=gh,e^c=ei,f^c=fh,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, 139859);