#9174286

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

4979 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=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, 9081482);