#289415

order	512
nilpotency class	3
isoclinism	isoclinic class of #289395
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#19629	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

619 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 619 conjugacy classes of subgroups — too many to draw. Only its 132 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 196611);