#3536220

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2368 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=fhi,d^2=fhi,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, 3443416);