#586708

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

1766 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)(6,9)(10,12)(13,14,18,15)(16,21,24,19)(17,20,25,22)(23,27,28,26)(29,30)(31,35)(32,34)(33,40)(36,39)(37,41)(38,43)(42,44);(7,11)(10,12)(14,15)(17,25)(19,21)(23,28)(29,31,34,39)(30,35,32,36)(33,38,41,44)(37,42,40,43);(5,6)(7,10)(8,9)(11,12)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,26,28,27)(29,32,34,30)(31,35,39,36)(33,40,41,37)(38,42,44,43);(5,7)(6,10)(8,11)(9,12)(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27)(29,31)(30,36)(32,35)(33,38)(34,39)(37,43)(40,42)(41,44);(5,6)(7,10)(8,9)(11,12)(13,17)(14,20)(15,22)(16,23)(18,25)(19,26)(21,27)(24,28)(29,33,34,41)(30,37,32,40)(31,38,39,44)(35,42,36,43)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=h,d^2=f,e^2=fh,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, 493904);