#144431

order	512
nilpotency class	4
isoclinism	isoclinic class of #144390
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	#3485	C₂²
LCS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 582 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)(17,18,22,27)(19,28,31,43)(20,23,32,25)(21,35,37,26)(24,36,40,30)(29,44,33,45)(34,42,47,39)(38,46,41,48);(1,3,5,10)(2,6,8,13)(4,9,11,16)(7,14,15,12)(18,23)(20,32)(24,38)(25,27)(26,39)(28,44)(29,33)(30,36)(34,47)(35,42)(40,41)(43,45);(1,4)(2,7)(3,9)(5,11)(6,12)(8,15)(10,16)(13,14)(17,19)(18,24)(20,33)(21,36)(22,31)(23,38)(25,41)(26,43)(27,40)(28,35)(29,32)(30,37)(34,46)(39,44)(42,45)(47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=i,e^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,d^c=dg,e^c=ei,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, 51627);