#743537

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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2136 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)(11,14,15,16)(18,19)(22,23)(25,26,29,31)(27,30,32,28);(10,12)(14,16)(17,18,21,19)(20,23,24,22)(25,27)(26,28)(29,32)(30,31);(11,15)(14,16)(18,19)(22,23)(25,27,29,32)(26,28,31,30);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,14,12,16)(17,19,21,18)(20,23,24,22)(25,27)(26,30)(28,31)(29,32);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,16,15,14)(17,20)(18,22)(19,23)(21,24)(25,28,29,30)(26,27,31,32)

Polycyclic presentation:

2^9|a^2=fgh,b^2=gh,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, 650733);