#489020

order	512
nilpotency class	4
isoclinism	isoclinic class of #488916
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#19189	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

709 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 709 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=hi,d^2=i,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fhi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fhi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396216);