#9017041

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4841 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8924237);