#149029

order	512
nilpotency class	4
isoclinism	isoclinic class of #13429
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2044
commutator	C₂×#41	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 615 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=d,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=dhi,f^b=fhi,g^b=ghi,h^b=hi,e^c=ehi,f^c=fhi,g^c=ghi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,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, 56225);