#488962

order	512
nilpotency class	4
isoclinism	isoclinic class of #488901
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	#61086	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 899 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) = 256

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

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

Polycyclic presentation:

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

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, 396158);