#110253

order	512
nilpotency class	3
isoclinism	isoclinic class of #110242
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

751 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 751 conjugacy classes of subgroups — too many to draw. Only its 150 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efi,d^2=hi,e^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=dh,e^b=ei,f^b=fi,d^c=di,e^c=ehi,f^c=fhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17449);