#153375

order	512
nilpotency class	5
isoclinism	isoclinic class of #153255
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2233
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂⁴	SD₃₂
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

194 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

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

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, 60571);