#3334945

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2714	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2723 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3242141);