#144427

order	512
nilpotency class	4
isoclinism	isoclinic class of #144375
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	C₂²×#41	C₂³
prime radical	#3487	C₂²
LCS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

565 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 565 conjugacy classes of subgroups — too many to draw. Only its 56 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)(3,6,10,13)(4,7,11,15)(9,12,16,14)(17,18,23,28)(19,29,32,44)(20,24,22,26)(21,34,36,27)(25,35,42,31)(30,46,37,45)(33,43,38,40)(39,48,41,47);(1,3,5,10)(2,6,8,13)(4,9,11,16)(7,14,15,12)(18,24)(19,30)(20,22)(26,28)(27,40)(29,44)(31,48)(32,37)(33,38)(34,43)(35,47)(39,41);(1,4)(2,7)(3,9)(5,11)(6,12)(8,15)(10,16)(13,14)(17,19)(18,25)(20,30)(21,35)(22,37)(23,32)(24,39)(26,41)(27,44)(28,42)(29,34)(31,36)(33,48)(38,47)(40,45)(43,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fghi,c^b=cf,d^b=di,e^b=eh,f^b=fi,d^c=dg,e^c=ei,f^c=fi,h^c=hi,f^e=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, 51623);