#3399365

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1943 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)(10,11)(12,16)(19,23)(21,24)(25,26,29,30)(27,28,31,32);(1,3,6,4)(2,7,5,8)(9,10,13,11)(12,14,16,15)(17,18,20,22)(19,21,23,24)(25,27,29,31)(26,28,30,32);(2,5)(7,8)(9,10,13,11)(12,15,16,14)(17,19)(18,21)(20,23)(22,24)(25,28)(26,27)(29,32)(30,31);(1,4,6,3)(2,7,5,8)(9,11,13,10)(12,14,16,15)(17,18)(19,21)(20,22)(23,24)(25,27)(26,28)(29,31)(30,32);(1,5,6,2)(3,8,4,7)(9,12,13,16)(10,14,11,15)(17,19)(18,21)(20,23)(22,24)(25,26)(27,28)(29,30)(31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=f,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, 3306561);