#144417

order	512
nilpotency class	4
isoclinism	isoclinic class of #144369
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

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

normal subgroups only

This group has 724 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)(10,15,18,29)(11,19)(12,22)(13,24)(16,30,33,40)(17,34,25,28)(20,37)(21,26)(27,35,31,32)(36,38);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,15)(13,25)(14,18)(17,24)(19,35)(20,31)(21,33)(22,29)(23,34)(26,32)(27,38)(28,39)(30,36)(37,40);(5,8)(9,11,14,21)(10,16,18,33)(12,20,22,38)(13,26,24,19)(15,27,29,31)(17,35,25,32)(23,37,39,36)(28,40,34,30)

Polycyclic presentation:

2^9|a^2=g,c^2=i,e^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fghi,c^b=cf,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, 51613);