#153361

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	C₂²×D₁₆	C₂²
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

221 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 221 conjugacy classes of subgroups — too many to draw. Only its 68 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)(5,6,8,13,11,15,21,10)(7,16,19,30,20,28,18,12)(9,23,22,24,26,14,32,31)(17,34,33,29,36,27,35,25);(6,10)(7,17)(8,21)(9,24)(12,27)(13,15)(14,32)(16,29)(18,33)(19,35)(20,36)(22,23)(25,28)(26,31)(30,34);(2,4)(5,7,8,18,11,20,21,19)(6,12,13,28,15,30,10,16)(9,25,22,27,26,29,32,34)(14,33,31,17,23,35,24,36)

Polycyclic presentation:

2^9|a^2=e,c^2=d,d^2=i,e^2=i,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fg,g^b=gh,h^b=hi,e^c=ei,f^c=fgi,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, 60557);