#4422102

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2154 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fh,c^2=g,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,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, 4329298);