#4310655

order	512
nilpotency class	2
isoclinism	isoclinic class of #4308740
automorphism	Aut(G) order 33554432
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2442 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=g,d^2=fi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=df

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, 4217851);