#5228750

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

1612 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=ghi,c^2=fg,d^2=f,e^2=f,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, 5135946);