#151839

order	512
nilpotency class	5
isoclinism	isoclinic class of #13584
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#353
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄⋊Q₈	C₂³
prime radical	C₂⁴	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂×C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 632 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bd,d^a=dg,f^a=fhi,g^a=gh,h^a=hi,c^b=ci,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, 59035);