#5465194

order	512
nilpotency class	2
isoclinism	isoclinic class of #5211343
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

1634 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fh,c^2=ghi,d^2=f,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,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, 5372390);