#550413

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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

1710 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)(5,6,9,8)(7,10,12,11)(13,14)(15,18)(16,19)(17,22)(20,25)(21,23)(24,27)(26,28)(29,30,34,35)(31,32,36,40)(33,41,42,37)(38,44,43,39);(1,2,4,3)(5,6,9,8)(7,11,12,10)(14,15)(16,23)(20,22)(24,28)(30,35)(31,36)(37,41)(38,43);(1,2,4,3)(5,7,9,12)(6,10,8,11)(13,15,18,14)(16,19,23,21)(17,22,25,20)(24,26,28,27)(29,31)(30,32)(33,38)(34,36)(35,40)(37,39)(41,44)(42,43);(5,6,9,8)(7,10,12,11)(13,16,18,23)(14,19,15,21)(17,24,25,28)(20,26,22,27)(29,32)(30,36)(31,35)(33,39)(34,40)(37,43)(38,41)(42,44);(1,3,4,2)(5,8,9,6)(7,11,12,10)(13,17,18,25)(14,20,15,22)(16,24,23,28)(19,26,21,27)(29,33,34,42)(30,37,35,41)(31,38,36,43)(32,39,40,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=fi,d^2=fgh,e^2=gi,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 457609);