#3335395

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2338 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fgh,c^2=f,e^2=fg,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, 3242591);