#5954454

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1977 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)(3,7)(9,10)(11,14)(12,13)(15,16)(17,18,21,23)(19,24,22,20)(27,33)(28,35)(31,36)(32,37);(1,2)(3,6)(4,5)(7,8)(9,11,13,16)(10,14,12,15)(17,18)(19,20)(21,23)(22,24)(26,30)(31,36)(32,37)(38,40);(3,7)(6,8)(10,12)(14,15)(17,19,21,22)(18,20,23,24)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,3)(2,6)(4,7)(5,8)(9,12,13,10)(11,15,16,14)(17,18)(19,20)(21,23)(22,24)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(1,2)(3,8)(4,5)(6,7)(9,11)(10,14)(12,15)(13,16)(17,20)(18,22)(19,23)(21,24)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)

Polycyclic presentation:

2^9|a^2=fghi,b^2=fhi,c^2=fghi,d^2=fhi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=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, 5861650);