#144234

order	512
nilpotency class	4
isoclinism	isoclinic class of #144232
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	#32933	C₂
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 609 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) = 64

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

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

Polycyclic presentation:

2^9|a^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=fh,g^a=gi,c^b=cf,d^b=dg,e^b=eh,g^b=gi,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, 51430);