#4648927

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1728 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)(3,4)(5,7)(6,8)(10,13)(11,15)(17,18)(19,23)(20,24)(21,22)(25,26,29,32)(27,33,34,30)(28,31,36,38)(35,39,40,37);(1,3)(2,6)(4,7)(5,8)(9,10,12,13)(11,14,15,16)(17,19,21,24)(18,20,22,23)(27,34)(30,33)(35,40)(37,39);(2,7)(3,8)(9,11,12,15)(10,14,13,16)(17,18,21,22)(19,23,24,20)(25,26)(27,30)(28,31)(29,32)(33,34)(35,37)(36,38)(39,40);(1,4)(2,3)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,19)(21,23)(22,24)(25,27,29,34)(26,30,32,33)(28,35,36,40)(31,37,38,39);(1,3)(2,4)(5,8)(6,7)(9,11)(10,14)(12,15)(13,16)(17,18)(19,20)(21,22)(23,24)(25,28,29,36)(26,31,32,38)(27,35,34,40)(30,37,33,39)

Polycyclic presentation:

2^9|a^2=fg,b^2=fgi,c^2=fgi,d^2=fg,e^2=fg,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, 4556123);