#9122471

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	#59029	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4830 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9029667);