#233335

order	512
nilpotency class	3
isoclinism	isoclinic class of #233331
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×#3253	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1336 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 1336 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) = 128

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

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

Polycyclic presentation:

2^9|c^2=i,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140531);