#144883

order	512
nilpotency class	4
isoclinism	isoclinic class of #142394
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1613
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	#30253	C₂
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 466 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) = 64

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

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

Polycyclic presentation:

2^9|c^2=d,d^2=i,e^2=gi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fgi,g^b=gi,h^b=hi,e^c=egi,f^c=fg,g^c=gi,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, 52079);