#9061583

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

5306 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)(11,16)(12,15)(13,14)(17,18,23,28)(19,24,27,22)(20,21,26,25)(29,32,31,30);(1,2,5,6)(3,7,8,4)(9,11)(10,12)(13,16)(14,15)(17,19,23,27)(18,24,28,22)(20,31,26,29)(21,30,25,32);(1,2)(3,7)(4,8)(5,6)(10,13)(12,16)(17,20,23,26)(18,25,28,21)(19,29,27,31)(22,32,24,30);(2,6)(3,8)(9,10,14,13)(11,16,15,12)(17,20)(18,21)(19,29)(22,30)(23,26)(24,32)(25,28)(27,31);(1,3)(2,4)(5,8)(6,7)(9,12,14,16)(10,15,13,11)(17,21)(18,26)(19,30)(20,28)(22,31)(23,25)(24,29)(27,32);(1,4)(2,3)(5,7)(6,8)(9,13,14,10)(11,16,15,12)(17,22,23,24)(18,27,28,19)(20,30,26,32)(21,31,25,29)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=g,e^2=g,f^2=h,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, 8968779);