#686050

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

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

2097 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=h,c^2=h,d^2=fg,e^2=fhi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 593246);