#144411

order	512
nilpotency class	4
isoclinism	isoclinic class of #144375
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

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

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

normal subgroups only

This group has 696 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) = 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)(10,16,19,25)(11,20)(13,14)(17,33,35,38)(18,32,28,31)(21,30,36,34)(22,37)(23,24)(26,29)(39,40);(1,3)(2,5)(4,7)(6,8)(9,10)(11,21)(12,25)(13,18)(14,28)(15,19)(16,27)(17,23)(20,35)(22,33)(24,36)(26,32)(29,31)(30,40)(34,37)(38,39);(5,8)(9,11,15,24)(10,17,19,35)(12,22,27,39)(13,23,14,20)(16,30,25,34)(18,36,28,21)(26,40,29,37)(31,38,32,33)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fghi,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dg,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, 51607);