#4130940

order	512
nilpotency class	2
isoclinism	isoclinic class of #4122169
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

1698 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)(3,5)(4,6)(7,8)(9,10,12,13)(11,14,16,15)(17,18,20,23)(19,22,24,21)(27,31)(29,32);(2,3)(4,8)(10,13)(14,15)(17,19,20,24)(18,21,23,22);(1,2,5,3)(4,7,8,6)(9,11,12,16)(10,14,13,15)(17,18,20,23)(19,22,24,21)(25,26,28,30)(27,29,31,32);(1,3,5,2)(4,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,24)(21,23)(25,27)(26,29)(28,31)(30,32);(1,4,5,8)(2,6,3,7)(9,11)(10,15)(12,16)(13,14)(17,18)(19,21)(20,23)(22,24)

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=ghi,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4038136);