#5250596

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

1990 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)(3,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,22)(21,24)(25,26,29,30)(27,31,33,37)(28,32,35,36)(34,38,39,40);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,13,14,16)(17,18)(19,23)(20,22)(21,24)(26,30)(27,33)(28,35)(38,40);(2,7)(3,8)(10,14)(13,16)(17,18,20,22)(19,21,24,23)(25,26)(27,31)(28,36)(29,30)(32,35)(33,37)(34,40)(38,39);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,21)(20,22)(23,24)(25,27)(26,31)(28,34)(29,33)(30,37)(32,38)(35,39)(36,40);(1,4)(2,3)(5,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,21)(20,24)(22,23)(25,28)(26,32)(27,34)(29,35)(30,36)(31,38)(33,39)(37,40)

Polycyclic presentation:

2^9|a^2=f,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, 5157792);