#143111

order	512
nilpotency class	4
isoclinism	isoclinic class of #143106
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	#3169	C₂²
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

352 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

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

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, 50307);