#606473

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

1634 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)(6,7)(8,12)(15,21)(19,24)(22,27)(26,28)(29,30,35,33)(31,37,32,36)(34,41,42,38)(39,43,40,44);(6,7)(10,11)(14,18)(19,24)(20,25)(26,28)(29,31,35,32)(30,36,33,37)(34,40,42,39)(38,44,41,43);(1,2,4,3)(5,6,9,7)(8,10,12,11)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,32,35,31)(30,37,33,36)(34,40,42,39)(38,44,41,43);(1,2,4,3)(5,7,9,6)(8,11,12,10)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,33,35,30)(31,36,32,37)(34,41,42,38)(39,43,40,44);(1,3,4,2)(5,8)(6,10)(7,11)(9,12)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,34,35,42)(30,38,33,41)(31,39,32,40)(36,43,37,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=hi,d^2=fgi,e^2=fghi,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, 513669);