#144239

order	512
nilpotency class	4
isoclinism	isoclinic class of #144236
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

580 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=gi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,e^b=eh,g^b=gi,d^c=di,e^c=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51435);