#143249

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

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 ◁ C₂×#446 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

459 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 459 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

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

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, 50445);