#5956374

order	512
nilpotency class	2
isoclinism	isoclinic class of #5955303
automorphism	Aut(G) order 2097152
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

1842 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,c^2=h,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5863570);