#143545

order	512
nilpotency class	4
isoclinism	isoclinic class of #143522
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#469
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

339 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 339 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(3,10)(4,5)(7,11)(8,14)(12,13)(15,16)(17,18,23,28)(19,29,31,48)(20,33,32,25)(21,36,39,42)(22,40,34,27)(24,41,43,30)(26,45,46,38)(35,44,37,47);(1,2)(3,11)(4,8)(5,14)(6,9)(7,13)(10,15)(12,16)(17,19,23,31)(18,24,28,43)(20,34,32,22)(21,37,39,35)(25,26,33,46)(27,47,40,44)(29,42,48,36)(30,45,41,38);(1,3)(2,7)(4,13)(5,10)(6,12)(8,16)(9,15)(11,14)(17,20,22,19,23,32,34,31)(18,25,27,29,28,33,40,48)(21,35,38,30,39,37,45,41)(24,42,44,26,43,36,47,46)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50741);