#143240

order	512
nilpotency class	4
isoclinism	isoclinic class of #143214
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1770 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

517 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 517 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50436);