#151875

order	512
nilpotency class	5
isoclinism	isoclinic class of #13605
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×#124	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 563 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59071);