#3463902

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
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)(9,10,14,15)(11,16,20,23)(12,21,22,17)(13,18,24,19)(25,26)(27,32)(28,35)(29,31)(30,37)(33,38)(34,40)(36,39);(6,8)(10,15)(12,22)(16,23)(18,19)(25,27,31,28)(26,32,29,35)(30,34,38,36)(33,39,37,40);(5,6)(7,8)(9,11)(10,16)(12,19)(13,21)(14,20)(15,23)(17,24)(18,22)(25,28,31,27)(26,32,29,35)(30,36,38,34)(33,39,37,40);(1,2,3,4)(9,12,14,22)(10,17,15,21)(11,19,20,18)(13,16,24,23)(25,29,31,26)(27,32,28,35)(30,37,38,33)(34,39,36,40);(9,13,14,24)(10,18,15,19)(11,17,20,21)(12,23,22,16)(25,30)(26,33)(27,34)(28,36)(29,37)(31,38)(32,39)(35,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3371098);