#153327

order	512
nilpotency class	5
isoclinism	isoclinic class of #153213
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2226
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#509
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

228 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=d,c^2=dh,e^2=ghi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fi,g^a=gh,h^a=hi,c^b=cf,d^c=di,e^c=eg,f^c=fi,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224983, #72225004, #72225008
maximal normal subgroup of	#72224983, #72225004, #72225008

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60523);