#104954

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#394
commutator, Frattini, cosocle	C₄³	C₂³
prime radical	C₂×#127	D₄
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 385 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,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, 12150);