#585356

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

1770 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)(7,11)(10,12)(13,14)(15,18)(16,21)(17,22)(19,23)(20,25)(24,26)(27,28)(29,30,33,36)(31,37,39,34)(32,40,41,35)(38,42,44,43);(6,9)(10,12)(14,15)(16,23)(17,25)(26,27)(29,30,33,36)(31,37,39,34)(32,35,41,40)(38,43,44,42);(5,6)(7,10)(8,9)(11,12)(13,15,18,14)(16,19,23,21)(17,22,25,20)(24,26,28,27)(29,30)(31,34)(32,35)(33,36)(37,39)(38,42)(40,41)(43,44);(5,7)(6,10)(8,11)(9,12)(13,16,18,23)(14,19,15,21)(17,24,25,28)(20,26,22,27)(29,31,33,39)(30,34,36,37)(32,38,41,44)(35,42,40,43);(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)(29,32,33,41)(30,35,36,40)(31,38,39,44)(34,42,37,43)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=f,d^2=g,e^2=fg,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, 492552);