#484628

order	512
nilpotency class	4
isoclinism	isoclinic class of #484199
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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 ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

933 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:

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

Polycyclic presentation:

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

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, 391824);