#142598

order	512
nilpotency class	4
isoclinism	isoclinic class of #142548
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×#117	C₂³
prime radical	#3493	C₂²
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

461 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 461 conjugacy classes of subgroups — too many to draw. Only its 54 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=g,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,f^b=fi,h^b=hi,d^c=dh,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49794);