#146856

order	512
nilpotency class	4
isoclinism	isoclinic class of #11499
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂⁴	C₂×C₄²
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂²	C₄×#99
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

680 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,g^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,f^c=fh,f^e=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54052);