#4032176

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

2564 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fhi,c^2=i,d^2=ghi,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, 3939372);