#104980

order	512
nilpotency class	3
isoclinism	isoclinic class of #104973
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#394
commutator, Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

410 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=d,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,c^b=cf,d^b=dg,e^b=eg,f^b=fi,e^c=egh,f^c=fgi

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, 12176);