#4033826

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1504 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=fghi,d^2=fhi,e^2=h,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, 3941022);