#9105302

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

4757 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=gh,e^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=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, 9012498);