#144369

order	512
nilpotency class	4
isoclinism	isoclinic class of #144369
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1001 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 1001 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)(11,19)(12,16)(13,17)(14,18)(15,21)(20,35)(22,34)(23,33)(24,30)(25,31)(26,32)(27,36)(28,37)(29,38)(39,40);(5,8)(9,11)(10,15)(12,21)(13,20)(14,22)(16,19)(17,27)(18,29)(23,26)(24,38)(25,36)(28,32)(30,34)(31,35)(33,40)(37,39);(1,3)(2,5)(4,7)(6,8)(11,20)(13,26)(15,27)(16,30)(17,32)(19,33)(21,36)(22,23)(25,39)(28,29)(34,35)(37,38)

Polycyclic presentation:

2^9|f^2=i,b^a=bd,c^a=ce,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,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, 51565);