#153384

order	512
nilpotency class	5
isoclinism	isoclinic class of #153255
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2233
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	C₂×SD₃₂
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 175 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) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=d,c^2=dh,f^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,f^b=fg,g^b=gh,h^b=hi,d^c=di,f^c=fg,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, 60580);