#151850

order	512
nilpotency class	5
isoclinism	isoclinic class of #1412
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#354
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄⋊Q₈	C₂³
socle	C₂²	C₂×#354
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂×C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

648 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bd,d^a=dg,f^a=fh,g^a=gh,h^a=hi,d^b=di,e^b=egh,e^d=ehi,f^e=fi,g^e=gi

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, 59046);