#4500243

order	512
nilpotency class	2
isoclinism	isoclinic class of #4465744
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1614 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=h,d^2=i,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4407439);