#9133770

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

5060 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9040966);