#3553704

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2059 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fh,c^2=hi,d^2=fhi,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, 3460900);