#488923

order	512
nilpotency class	4
isoclinism	isoclinic class of #488920
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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 ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

994 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 156 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:

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

Polycyclic presentation:

2^9|b^2=i,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, 396119);