#5034877

order	512
nilpotency class	2
isoclinism	isoclinic class of #5024635
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×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4942073);