#591869

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)(6,7)(8,12)(13,14,17,18)(15,21,22,19)(16,24,25,20)(23,26,28,27)(31,37)(34,40)(38,43)(41,44);(6,7)(10,11)(14,18)(15,22)(16,25)(26,27)(29,30,33,36)(31,34,37,40)(32,35,39,42)(38,41,43,44);(1,3,4,2)(5,6,9,7)(8,10,12,11)(13,15,17,22)(14,19,18,21)(16,23,25,28)(20,26,24,27)(29,30)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,44);(5,7,9,6)(8,11,12,10)(13,14)(15,21)(16,20)(17,18)(19,22)(23,27)(24,25)(26,28)(29,31)(30,34)(32,38)(33,37)(35,41)(36,40)(39,43)(42,44);(5,8,9,12)(6,10,7,11)(13,16,17,25)(14,20,18,24)(15,23,22,28)(19,26,21,27)(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=fg,e^2=h,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, 499065);