#144724

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	#29745	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

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

normal subgroups only

This group has 460 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)(7,16)(8,21)(9,10)(12,27)(13,29)(14,25)(17,30)(18,34)(19,35)(20,23)(22,31)(24,26)(28,36)(32,33);(1,2)(3,4)(5,6)(7,17)(8,22)(9,25)(10,14)(11,15)(12,19)(13,24)(16,33)(18,32)(20,28)(21,31)(23,27)(26,29)(30,34)(35,36);(1,3,4,2)(5,7,10,19,11,18,9,20)(6,12,14,32,15,28,25,17)(8,23,21,34,24,35,26,16)(13,30,29,36,22,33,31,27)

Polycyclic presentation:

2^9|c^2=dg,e^2=gi,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gi,c^b=cf,e^b=eh,f^b=fi,d^c=di,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, 51920);