#3967510

order	512
nilpotency class	2
isoclinism	isoclinic class of #3858029
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2618 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ghi,d^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3874706);