#144391

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

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

normal subgroups only

This group has 894 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)(11,16)(12,15)(13,25)(14,19)(17,23)(18,26)(20,35)(21,33)(22,32)(24,34)(27,31)(28,38)(29,39)(30,36)(37,40);(5,8)(10,15)(11,20)(12,23)(13,26)(16,30)(17,19)(18,34)(21,37)(22,27)(25,29)(28,31)(32,40)(33,35)(36,38);(1,3)(2,5)(4,7)(6,8)(9,11,14,22)(10,16,19,32)(12,21,23,38)(13,27,26,20)(15,28,17,33)(18,35,25,31)(24,37,39,36)(29,40,34,30)

Polycyclic presentation:

2^9|c^2=i,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fi,d^c=dg,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, 51587);