#2607335

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2216 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)(10,14)(12,19)(15,20)(18,23)(25,26)(27,31)(28,32)(29,30)(33,39)(34,37)(35,40)(36,38);(5,6,7,8)(10,14)(11,17)(16,21)(18,23)(26,30)(31,37)(32,38)(39,40);(5,6)(7,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,26)(27,31)(28,32)(29,30)(33,39)(34,37)(35,40)(36,38);(1,2)(3,4)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,27,29,34)(26,31,30,37)(28,35,36,33)(32,40,38,39);(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,28,29,36)(26,32,30,38)(27,33,34,35)(31,39,37,40)

Polycyclic presentation:

2^9|a^2=i,b^2=fgh,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2514531);