#4632831

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 2097152
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

1532 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)(3,4)(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(19,23)(21,24)(25,26,30,27)(28,31,36,33)(29,32,37,34)(35,38,40,39);(1,2,5,6)(3,7,8,4)(9,11,13,16)(10,12,14,15)(17,18,20,22)(19,21,23,24)(26,27)(31,33)(32,34)(38,39);(2,6)(4,7)(9,10,13,14)(11,15,16,12)(17,18)(19,21)(20,22)(23,24)(25,27,30,26)(28,33,36,31)(29,32,37,34)(35,38,40,39);(1,3)(2,4)(5,8)(6,7)(9,12)(10,11)(13,15)(14,16)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(1,4)(2,3)(5,7)(6,8)(9,10)(11,12)(13,14)(15,16)(17,19)(18,21)(20,23)(22,24)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=fg,b^2=fgh,c^2=fgi,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,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, 4540027);