#140547

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×C₂²⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

366 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=eg,d^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=cgi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47743);