#10086377

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

4901 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gi,d^2=ghi,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9993573);