#4720159

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1578 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=fghi,d^2=fhi,e^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, 4627355);