#5140958

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
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

1752 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)(10,13)(14,16)(17,18)(19,23)(20,22)(21,24)(25,26,30,33)(27,28,35,31)(29,37,39,32)(34,40,38,36);(11,15)(14,16)(17,18)(19,23)(20,22)(21,24)(25,27,30,35)(26,31,33,28)(29,38,39,34)(32,36,37,40);(9,10,12,13)(11,14,15,16)(17,18,20,22)(19,21,24,23)(25,27)(26,31)(28,33)(29,38)(30,35)(32,36)(34,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24)(25,28,30,31)(26,27,33,35)(29,36,39,40)(32,34,37,38);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,21)(20,24)(22,23)(25,29,30,39)(26,32,33,37)(27,34,35,38)(28,36,31,40)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fg,d^2=f,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 5048154);