#3337154

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

2056 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)(9,10,13,14)(11,15,18,22)(12,16,21,20)(17,23,19,24)(25,26,30,28)(27,33,35,31)(29,36,38,32)(34,39,37,40);(6,8)(10,14)(15,22)(16,20)(23,24)(25,27,30,35)(26,31,28,33)(29,34,38,37)(32,39,36,40);(5,6,7,8)(9,11,13,18)(10,15,14,22)(12,19,21,17)(16,24,20,23)(25,27)(26,31)(28,33)(29,37)(30,35)(32,40)(34,38)(36,39);(1,2)(3,4)(9,10)(11,15)(12,20)(13,14)(16,21)(17,24)(18,22)(19,23)(25,28,30,26)(27,33,35,31)(29,32,38,36)(34,39,37,40);(9,12,13,21)(10,16,14,20)(11,17,18,19)(15,23,22,24)(25,29,30,38)(26,32,28,36)(27,34,35,37)(31,39,33,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=g,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, 3244350);