#743538

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1958 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)(11,15)(12,14)(13,16)(17,18,21,22)(19,23,24,20)(26,30)(27,32);(9,10,12,14)(11,13,16,15)(18,22)(20,23)(25,26,29,30)(27,28,32,31);(11,16)(13,15)(17,19,21,24)(18,20,22,23)(25,27,29,32)(26,31,30,28);(1,2)(3,6)(4,5)(7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,22)(21,24)(25,27)(26,28)(29,32)(30,31);(1,3,4,7)(2,6,5,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,23)(18,19,22,24)(25,28)(26,27)(29,31)(30,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=fgh,c^2=g,d^2=fg,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, 650734);