#571365

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=ghi,e^2=fi,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, 478561);