#104957

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

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

normal subgroups only

This group has 353 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=di,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, 12153);