#406728

order	512
nilpotency class	3
isoclinism	isoclinic class of #394470
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1268 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=f,d^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313924);