#141006

order	512
nilpotency class	4
isoclinism	isoclinic class of #125143
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#333
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

334 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 122 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=ei,d^2=ghi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,f^b=fhi,h^b=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48202);