#143108

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	#28174	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 390 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)(3,11)(4,9)(5,8)(6,10)(7,12)(13,16)(14,15)(17,18,23,29)(19,30,34,43)(20,24,37,40)(21,28,22,27)(25,31,44,33)(26,36,39,35)(32,42,47,41)(38,46,45,48);(1,3)(2,7)(4,14)(5,12)(6,13)(8,16)(9,11)(10,15)(18,24)(19,31)(20,35)(21,22)(25,41)(26,27)(28,39)(29,40)(30,45)(32,47)(33,34)(36,37)(38,43)(42,44);(2,8)(4,5)(7,11)(9,10)(12,14)(15,16)(17,19)(18,25)(20,32)(21,31)(22,33)(23,34)(24,38)(26,42)(27,43)(28,30)(29,44)(35,46)(36,48)(37,47)(39,41)(40,45)

Polycyclic presentation:

2^9|a^2=i,d^2=gi,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,e^b=ei,f^b=fi,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, 50304);