#5602187

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1478 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=fh,d^2=h,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef

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, 5509383);