#489017

order	512
nilpotency class	4
isoclinism	isoclinic class of #488916
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
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

738 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|b^2=h,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396213);