#119920

order	512
nilpotency class	3
isoclinism	isoclinic class of #119915
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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

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

normal subgroups only

This group has 700 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27116);