#5035252

order	512
nilpotency class	2
isoclinism	isoclinic class of #5024635
automorphism	Aut(G) order 6291456
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

3113 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=fh,d^2=i,e^2=f,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, 4942448);