#151846

order	512
nilpotency class	5
isoclinism	isoclinic class of #1412
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#354
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×#127	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×#127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 664 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)(7,16,19,25)(8,12)(9,14,22,13)(10,15)(11,18,24,32)(17,29,31,35)(20,27,33,26)(21,28)(23,30,34,36);(1,2,3,4)(5,7,9,19,10,18,22,32)(6,11,14,16,15,25,13,24)(8,17,20,31,21,30,33,36)(12,23,27,29,28,35,26,34);(5,8,10,21)(6,12,15,28)(7,17,18,30)(9,20,22,33)(11,23,25,35)(13,26,14,27)(16,29,24,34)(19,31,32,36)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=hi,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, 59042);