#144376

order	512
nilpotency class	4
isoclinism	isoclinic class of #144375
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂²×#41	C₂³
prime radical	#23915	C₂
LCS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 883 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)(12,25)(13,16)(14,19)(15,22)(17,29)(18,30)(21,38)(23,37)(24,34)(26,32)(27,33)(28,35)(31,39)(36,40);(5,8)(9,11,14,24)(10,15,19,32)(12,26,29,22)(13,21,30,23)(16,31,18,36)(17,34,25,20)(27,39,28,40)(33,37,35,38);(1,3)(2,5)(4,7)(6,8)(10,16)(11,21)(13,30)(17,35)(18,19)(20,34)(22,39)(23,24)(25,33)(26,40)(27,28)(31,36)

Polycyclic presentation:

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

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, 51572);