#4915049

order	512
nilpotency class	2
isoclinism	isoclinic class of #4861280
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2019 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=f,d^2=f,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4822245);