#142722

order	512
nilpotency class	4
isoclinism	isoclinic class of #142640
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1752 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

414 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 68 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)(3,6)(4,7)(5,8)(9,12)(10,13)(11,15)(14,16)(17,18,22,26)(19,28,32,39)(20,24,36,42)(21,37,35,27)(23,30,40,29)(25,44,43,34)(31,48,38,45)(33,46,47,41);(1,3,5,10)(2,6,8,13)(4,9,11,16)(7,14,15,12)(17,19,22,32)(18,23,26,40)(20,33,36,47)(21,38,35,31)(24,28,42,39)(25,45,43,48)(27,46,37,41)(29,44,30,34);(1,4)(2,7)(3,9)(5,11)(6,12)(8,15)(10,16)(13,14)(18,24)(20,34)(21,35)(23,39)(25,27)(26,42)(28,40)(29,33)(30,47)(31,38)(36,44)(37,43)(41,45)(46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=gi,e^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fgh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=egi,g^b=gi,h^b=hi,d^c=dh,e^c=eg,g^c=gi,h^c=hi

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, 49918);