#5490509

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
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

1580 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fgi,c^2=fg,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=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, 5397705);