#593123

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

1638 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=fgh,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, 500319);