#233327

order	512
nilpotency class	3
isoclinism	isoclinic class of #233323
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1328 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1328 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) = 128

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

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

Polycyclic presentation:

2^9|d^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,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, 140523);