#9062182

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#60044	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4678 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

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, 8969378);