#144418

order	512
nilpotency class	4
isoclinism	isoclinic class of #144390
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	#23913	C₂
LCS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 761 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fghi,c^b=cf,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, 51614);