#758213

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
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	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2406 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=gh,c^2=g,d^2=fi,e^2=fi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=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, 665409);