#144758

order	512
nilpotency class	4
isoclinism	isoclinic class of #142406
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1616
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

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

normal subgroups only

This group has 371 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) = 64

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

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

Polycyclic presentation:

2^9|b^2=g,c^2=d,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=fhi,g^a=gi,c^b=cf,e^b=eh,e^c=egi,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, 51954);