#145386

order	512
nilpotency class	4
isoclinism	isoclinic class of #143943
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#459
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

336 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 44 points:

(5,6)(7,9)(8,10)(11,12)(13,14,20,26)(15,27,30,40)(16,21,28,42)(17,35,18,37)(19,39,36,25)(22,33,43,29)(23,31,24,32)(34,44,38,41);(1,2)(3,4)(6,9)(10,11)(13,15,20,30)(14,21,26,42)(16,31,28,32)(17,34,18,38)(19,33,36,29)(22,23,43,24)(25,44,39,41)(27,37,40,35);(2,4)(6,10)(8,12)(9,11)(13,16,32,15,19,34,18,29,20,28,31,30,36,38,17,33)(14,22,35,44,25,27,24,21,26,43,37,41,39,40,23,42)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=dh,d^2=gi,e^2=hi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,f^b=fg,g^b=gi,h^b=hi,e^c=eghi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52582);