#143546

order	512
nilpotency class	4
isoclinism	isoclinic class of #143518
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#33771	C₂
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

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

normal subgroups only

This group has 382 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) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50742);