#144795

order	512
nilpotency class	4
isoclinism	isoclinic class of #142310
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1598
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

358 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=dg,d^2=i,e^2=gi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fh,g^a=gi,c^b=cf,e^b=eh,d^c=di,e^c=eg,f^c=fi,g^c=gi,h^c=hi,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51991);