#9214263

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

5816 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:

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

Polycyclic presentation:

2^9|a^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,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, 9121459);