#4131496

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
automorphism	Aut(G) order 1048576
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2276 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=fg,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,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, 4038692);