#143358

order	512
nilpotency class	4
isoclinism	isoclinic class of #143212
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	C₂²×D₈	D₄
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

372 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50554);