#488740

order	512
nilpotency class	4
isoclinism	isoclinic class of #488699
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

819 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395936);