#4131220

order	512
nilpotency class	2
isoclinism	isoclinic class of #4122169
automorphism	Aut(G) order 6291456
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

1782 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)(9,10,14,12)(11,16,13,15)(17,18,20,21)(19,22,24,23)(25,26,28,30)(27,31,32,29);(2,5)(3,8)(9,11,14,13)(10,15,12,16)(18,21)(22,23)(27,32)(29,31);(1,3)(2,6)(4,8)(5,7)(9,10,14,12)(11,16,13,15)(17,19,20,24)(18,22,21,23)(25,27)(26,29)(28,32)(30,31);(1,3)(2,6)(4,8)(5,7)(9,12,14,10)(11,15,13,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,29)(28,30)(31,32);(1,2)(3,7)(4,5)(6,8)(9,13,14,11)(10,16,12,15)(17,19)(18,23)(20,24)(21,22)(25,27)(26,29)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=hi,d^2=gh,e^2=gh,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, 4038416);