#489202

order	512
nilpotency class	4
isoclinism	isoclinic class of #488920
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 ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

824 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 148 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)(12,20)(13,28)(14,18)(15,21)(16,24)(17,23)(19,30)(25,36)(26,35)(27,32)(29,38)(31,37)(33,40)(34,39);(3,7)(5,8)(9,11)(10,16)(12,23)(13,29)(14,26)(15,25)(17,32)(18,33)(19,35)(20,38)(21,34)(22,28)(24,30)(27,39)(31,40)(36,37);(1,2,4,6)(3,8,7,5)(9,11)(10,17)(12,26)(13,24)(14,23)(15,25)(16,32)(18,36)(19,34)(20,22)(21,35)(27,40)(28,38)(29,30)(31,39)(33,37);(9,12,15,14)(10,18,21,20)(11,23,25,26)(13,27,31,30)(16,33,34,38)(17,22,35,36)(19,37,32,28)(24,29,39,40)

Polycyclic presentation:

2^9|c^2=h,d^2=i,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396398);