#612423

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1874 conjugacy classes of subgroups; only the 610 normal subgroups are shown.

normal subgroups only

This group has 1874 conjugacy classes of subgroups — too many to draw. Only its 610 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) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=g,d^2=fh,e^2=fhi,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^c=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 519619);