#144409

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	#3485	C₂²
LCS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 707 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)(10,15,18,22)(11,19)(13,25)(16,30,33,40)(17,34,26,29)(20,36)(21,27)(23,38)(28,35,31,32)(37,39);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,22)(13,26)(14,18)(15,24)(17,25)(19,35)(20,28)(21,33)(23,29)(27,32)(30,39)(31,37)(34,38)(36,40);(5,8)(9,11,14,21)(10,16,18,33)(12,20,24,37)(13,27,25,19)(15,28,22,31)(17,35,26,32)(23,39,38,36)(29,40,34,30)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fghi,c^b=cf,d^b=di,e^b=eh,g^b=gi,d^c=dg,e^c=ei,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, 51605);