#698354

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1964 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=f,d^2=gh,e^2=f,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, 605550);