#112535

order	512
nilpotency class	3
isoclinism	isoclinic class of #112521
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1008 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 1008 conjugacy classes of subgroups — too many to draw. Only its 166 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=di,g^b=ghi,d^c=di,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, 19731);