#153268

order	512
nilpotency class	5
isoclinism	isoclinic class of #153105
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2206
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#2246	C₂
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

352 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|c^2=d,d^2=i,e^2=gh,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fg,g^a=gh,h^a=hi,c^b=cf,d^b=di,e^b=eg,f^b=fg,g^b=gh,h^b=hi,e^c=egi,f^c=fgi,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224983, #72225000, #72225012
maximal normal subgroup of	#72224983, #72225000, #72225012

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60464);