#488979

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	#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

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

normal subgroups only

This group has 898 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)(13,25)(17,32)(18,26)(19,30)(20,31)(22,24)(23,38)(27,28)(29,34)(33,40)(35,39);(2,6)(3,4)(7,8)(9,10,15,20)(11,17,24,35)(12,18,14,19)(13,26,28,30)(16,27,31,25)(21,34,22,33)(23,39,38,32)(29,37,40,36);(2,7)(3,4)(6,8)(9,11)(10,17)(12,21)(13,23)(14,22)(15,24)(16,29)(18,34)(19,33)(20,35)(25,36)(26,39)(27,37)(28,38)(30,32)(31,40);(9,12,15,14)(10,18,20,19)(11,22,24,21)(13,25,28,27)(16,30,31,26)(17,33,35,34)(23,37,38,36)(29,39,40,32)

Polycyclic presentation:

2^9|a^2=h,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, 396175);