#233343

order	512
nilpotency class	3
isoclinism	isoclinic class of #233341
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1272 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 1272 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)(3,10)(4,15)(5,9)(6,8)(7,12)(11,14)(13,16)(17,18)(19,29)(20,31)(21,28)(22,27)(23,26)(24,30)(25,32);(1,3,6,14)(2,7,8,16)(4,13,5,12)(9,11,15,10)(17,19,23,25)(18,24,26,20)(21,31,22,30)(27,32,28,29);(1,2)(3,11)(4,9)(5,15)(6,8)(7,12)(10,14)(13,16)(17,20,23,24)(18,25,26,19)(21,32,22,29)(27,31,28,30);(2,8)(7,16)(9,15)(10,11)(18,26)(20,24)(27,28)(29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140539);