#144878

order	512
nilpotency class	4
isoclinism	isoclinic class of #142256
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂×#469	C₂²
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 461 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)(5,6,12,18)(7,19,22,36)(8,23,9,27)(10,17,11,16)(13,30,32,20)(14,24,15,26)(21,34,29,33)(25,35,28,31);(1,2)(3,4)(7,20)(8,24)(9,26)(10,11)(13,19)(14,27)(15,23)(16,17)(21,28)(22,30)(25,29)(31,34)(32,36)(33,35);(1,3,4,2)(5,7)(6,13)(8,25)(9,28)(10,29)(11,21)(12,22)(14,33)(15,34)(16,35)(17,31)(18,32)(19,23)(20,26)(24,30)(27,36)

Polycyclic presentation:

2^9|a^2=i,c^2=d,e^2=gi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fgi,g^b=gi,h^b=hi,e^c=eg,f^c=fgi,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, 52074);