#104563

order	512
nilpotency class	3
isoclinism	isoclinic class of #104437
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#392
commutator, Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

449 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 449 conjugacy classes of subgroups — too many to draw. Only its 86 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)(10,14)(11,15)(12,16)(17,18,21,25)(19,26,29,27)(20,31,22,24)(23,30,32,28);(2,5)(3,8)(6,7)(9,10,13,11)(12,15,16,14)(17,19,21,29)(18,23,25,32)(20,30,22,28)(24,27,31,26);(3,8)(5,6)(9,11,13,10)(12,14,16,15)(17,18)(19,27)(20,24)(21,25)(22,31)(23,28)(26,29)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 11759);