#5250628

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	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1992 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=fhi,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, 5157824);