#488978

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
prime radical	#61084	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 946 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:

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

Polycyclic presentation:

2^9|a^2=hi,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, 396174);