#586410

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1618 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(14,18)(15,21)(16,23)(26,28)(31,37)(35,40)(38,43)(42,44);(5,6,8,10)(7,9,11,12)(14,18)(15,21)(20,25)(22,27)(30,34)(35,40)(36,41)(42,44);(5,7,8,11)(6,9,10,12)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,30)(31,35)(32,36)(33,34)(37,40)(38,42)(39,41)(43,44);(5,7)(6,9)(8,11)(10,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,31)(30,35)(32,38)(33,37)(34,40)(36,42)(39,43)(41,44);(5,6)(7,9)(8,10)(11,12)(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,32,33,39)(30,36,34,41)(31,38,37,43)(35,42,40,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=dh,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 493606);