#576047

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3358 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=fg,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, 483243);