#143116

order	512
nilpotency class	4
isoclinism	isoclinic class of #143102
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 ◁ C₂×#446 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 380 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) = 128

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

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=fh,g^a=gi,c^b=cf,d^b=dg,f^b=fi,g^b=gi,d^c=dh,e^c=ehi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50312);