#576026

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

3598 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=f,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, 483222);