#5491026

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
automorphism	Aut(G) order 2097152
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

1754 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)(3,5)(4,6)(7,8)(10,12)(14,15)(19,23)(22,24)(25,26,31,28)(27,34,29,32)(30,33,38,36)(35,40,37,39);(4,8)(6,7)(10,12)(14,15)(18,21)(22,24)(25,27,31,29)(26,32,28,34)(30,37,38,35)(33,40,36,39);(2,3)(6,7)(9,10,13,12)(11,15,16,14)(17,18)(19,22)(20,21)(23,24)(25,28,31,26)(27,32,29,34)(30,33,38,36)(35,40,37,39);(1,3,5,2)(4,7,8,6)(9,11,13,16)(10,14,12,15)(25,29,31,27)(26,32,28,34)(30,37,38,35)(33,39,36,40);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,15,16,14)(17,19)(18,22)(20,23)(21,24)(25,30,31,38)(26,33,28,36)(27,35,29,37)(32,39,34,40)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fgi,d^2=fg,e^2=fg,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, 5398222);