#599170

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

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)(6,9)(7,11)(13,14)(15,20)(16,22)(17,21)(18,19)(23,27)(24,28)(25,26)(31,37)(34,40)(38,43)(41,44);(6,9)(10,12)(14,19)(15,22)(17,25)(24,27)(29,30,33,36)(31,34,37,40)(32,35,39,42)(38,41,43,44);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15,18,22)(14,20,19,16)(17,23,25,28)(21,27,26,24)(29,30)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,44);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,16,18,20)(14,15,19,22)(17,24,25,27)(21,23,26,28)(29,31,33,37)(30,34,36,40)(32,38,39,43)(35,41,42,44);(5,7)(6,10)(8,11)(9,12)(13,17)(14,21)(15,23)(16,24)(18,25)(19,26)(20,27)(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=fgh,c^2=fi,d^2=ghi,e^2=fgh,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, 506366);