#2534504

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1648 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=gi,d^2=fgh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2441700);