#561861

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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

1694 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)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(29,30,33,34)(31,37,38,35)(32,40,41,36)(39,42,44,43);(1,3,4,2)(6,9)(10,12)(14,18)(16,24)(19,21)(22,28)(30,34)(31,38)(32,41)(42,43);(5,6)(7,10)(8,9)(11,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(29,31,33,38)(30,35,34,37)(32,39,41,44)(36,42,40,43);(5,7,8,11)(6,10,9,12)(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27)(29,30)(31,37)(32,36)(33,34)(35,38)(39,43)(40,41)(42,44);(13,16,17,24)(14,20,18,25)(15,22,23,28)(19,26,21,27)(29,32,33,41)(30,36,34,40)(31,39,38,44)(35,42,37,43)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=g,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^b=ef,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, 469057);