#119722

order	512
nilpotency class	3
isoclinism	isoclinic class of #119716
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

720 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,e^b=ehi,g^b=gh,e^c=ei,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26918);