#633889

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
automorphism	?
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

1754 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)(6,8)(9,10,12,13)(11,15,16,14)(17,18,21,24)(19,20,23,22)(26,29)(28,32);(10,13)(14,15)(25,26,30,29)(27,32,31,28);(17,19,21,23)(18,22,24,20)(25,27)(26,28)(29,32)(30,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24)(25,28,30,32)(26,31,29,27);(1,3,4,7)(2,6,5,8)(9,11,12,16)(10,14,13,15)(17,20,21,22)(18,23,24,19)(25,29,30,26)(27,32,31,28)

Polycyclic presentation:

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

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, 541085);