#5035971

order	512
nilpotency class	2
isoclinism	isoclinic class of #5024635
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2084 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=g,d^2=f,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=ef,e^d=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, 4943167);