#105334

order	512
nilpotency class	3
isoclinism	isoclinic class of #8101
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂³	C₄³
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

648 conjugacy classes of subgroups; only the 246 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 246 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) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 36 points:

(5,6,8,13,10,12,9,14)(7,15,16,20,17,11,18,19)(21,22,24,27,25,28,32,35)(23,29,31,36,30,26,33,34);(5,7,9,16,10,17,8,18)(6,11,14,19,12,15,13,20)(21,23)(22,26,28,29)(24,31)(25,30)(27,34,35,36)(32,33);(1,2,3,4)(6,12)(11,15)(13,14)(19,20)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,e^2=hi,f^2=h,b^a=bd,c^a=ch,d^a=dh,f^a=fi,d^b=di,e^b=eh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12530);