#4617130

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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

1928 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)(11,15)(12,16)(19,23)(22,24)(25,26)(27,29)(28,35)(30,31)(32,37)(33,38)(34,40)(36,39);(2,5)(6,8)(9,10,13,14)(11,16,15,12)(18,21)(22,24)(26,31)(27,33)(32,35)(34,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,15)(14,16)(17,18,20,21)(19,22,23,24)(25,27,30,33)(26,29,31,38)(28,34,37,39)(32,36,35,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,14)(15,16)(25,28,30,37)(26,32,31,35)(27,34,33,39)(29,36,38,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,16)(14,15)(17,19)(18,22)(20,23)(21,24)(25,29)(26,27)(28,36)(30,38)(31,33)(32,34)(35,39)(37,40)

Polycyclic presentation:

2^9|b^2=fgi,c^2=gh,d^2=f,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, 4524326);