#3973715

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=fgh,d^2=i,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, 3880911);