#489386

order	512
nilpotency class	4
isoclinism	isoclinic class of #489385
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
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

813 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 813 conjugacy classes of subgroups — too many to draw. Only its 136 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|d^2=gi,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fhi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,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, 396582);