#233344

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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1256 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1256 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) = 64

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

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

Polycyclic presentation:

2^9|d^2=hi,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, 140540);