#4127555

order	512
nilpotency class	2
isoclinism	isoclinic class of #4122169
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

1790 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)(10,13)(11,15)(17,18)(19,23)(20,22)(21,24)(25,26,31,30)(27,28,32,29);(2,4)(3,8)(9,10)(11,16)(12,13)(14,15)(17,18,20,22)(19,21,24,23)(25,27,31,32)(26,29,30,28);(2,4)(6,7)(10,13)(14,16)(17,18)(19,23)(20,22)(21,24)(25,28,31,29)(26,27,30,32);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,15)(13,16)(17,18)(19,23)(20,22)(21,24)(25,29,31,28)(26,32,30,27);(1,4,5,2)(3,6,8,7)(9,10)(11,16)(12,13)(14,15)(17,19)(18,21)(20,24)(22,23)(25,30,31,26)(27,29,32,28)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=fgh,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, 4034751);