#10040691

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

4511 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=hi,e^2=g,f^2=h,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, 9947887);