#4279666

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
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	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2050 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=gh,c^2=g,d^2=f,e^2=fhi,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, 4186862);