#719095

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	#15273	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2675 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fgi,c^2=fgi,d^2=fgh,e^2=fgh,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, 626291);