#593021

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

1642 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)(13,14)(15,21)(16,18)(17,24)(19,22)(20,25)(23,26)(27,28)(31,37)(34,40)(38,43)(41,44);(7,12)(9,11)(14,16)(15,22)(17,25)(26,27)(29,30,33,36)(31,34,37,40)(32,35,39,42)(38,41,43,44);(1,3,4,2)(5,7)(6,9)(8,12)(10,11)(13,15)(14,19)(16,21)(17,23)(18,22)(20,26)(24,27)(25,28)(29,30)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,44);(5,7)(6,9)(8,12)(10,11)(13,16,18,14)(15,19,22,21)(17,24,25,20)(23,26,28,27)(29,31,33,37)(30,34,36,40)(32,38,39,43)(35,41,42,44);(5,6)(7,9)(8,10)(11,12)(13,17)(14,20)(15,23)(16,24)(18,25)(19,26)(21,27)(22,28)(29,32,33,39)(30,35,36,42)(31,38,37,43)(34,41,40,44)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=fgi,d^2=fgh,e^2=gh,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, 500217);