#104972

order	512
nilpotency class	3
isoclinism	isoclinic class of #104937
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

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

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

normal subgroups only

This group has 335 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)(4,6,8,7)(10,13)(14,15)(18,23)(19,26)(20,21)(24,31)(25,29)(27,28);(2,5)(6,7)(9,10,11,13)(12,14,16,15)(17,18,22,25)(19,24,28,31)(20,29,21,23)(26,30,27,32);(2,6)(4,8)(5,7)(10,14)(12,16)(13,15)(17,19,21,26,22,28,20,27)(18,24,29,32,25,31,23,30)

Polycyclic presentation:

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

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, 12168);