#122161

order	512
nilpotency class	3
isoclinism	isoclinic class of #122159
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1446
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

690 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fh,e^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=di,g^b=gh,d^c=di,g^c=gh

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, 29357);