#584326

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
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

1738 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)(5,6,8,9)(7,11,12,10)(14,18)(19,24)(20,25)(26,28)(29,30,35,33)(31,38,32,36)(34,41,42,37)(39,43,40,44);(6,9)(10,11)(14,18)(15,21)(20,25)(22,27)(29,31,35,32)(30,36,33,38)(34,39,42,40)(37,43,41,44);(6,9)(10,11)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,32,35,31)(30,36,33,38)(34,40,42,39)(37,43,41,44);(5,6)(7,10)(8,9)(11,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,33,35,30)(31,38,32,36)(34,41,42,37)(39,44,40,43);(1,3,4,2)(5,7,8,12)(6,10,9,11)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,34)(30,37)(31,39)(32,40)(33,41)(35,42)(36,43)(38,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fgh,c^2=fgh,d^2=fg,e^2=fgi,b^a=bf,c^a=cg,d^a=dgh,e^a=ef,c^b=ch,d^b=dg,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, 491522);