#4414423

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
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

2058 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)(9,10,14,15)(11,12,17,20)(13,16,22,23)(18,19,21,24)(25,26)(27,28)(29,37)(30,33)(31,35)(32,38)(34,39)(36,40);(10,15)(11,17)(16,23)(18,21)(25,27,30,35)(26,28,33,31)(29,34,38,40)(32,36,37,39);(5,6,8,7)(9,11)(10,12)(13,21)(14,17)(15,20)(16,24)(18,22)(19,23)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,37)(38,40);(1,2)(3,4)(5,7,8,6)(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,23)(22,24)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,37,40);(9,13,14,22)(10,16,15,23)(11,18,17,21)(12,19,20,24)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,37)(35,40)

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=fgh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^c=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, 4321619);