#9027902

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

4740 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gi,d^2=g,e^2=g,f^2=i,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, 8935098);