#9040191

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

5853 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)(11,15)(12,16)(13,14)(17,18)(19,25)(20,21)(22,23)(24,26)(27,28)(29,31)(30,32);(1,2,4,6)(3,5,7,8)(9,10,13,14)(11,15,16,12)(18,23)(19,26)(20,29)(28,32);(1,3)(2,5)(4,7)(6,8)(9,11)(10,12)(13,16)(14,15)(20,29)(21,31)(27,30)(28,32);(9,12)(10,11)(13,15)(14,16)(17,19)(18,24)(20,30)(21,28)(22,26)(23,25)(27,29)(31,32);(1,2)(3,5)(4,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,20)(18,21)(19,27)(22,29)(23,31)(24,32)(25,28)(26,30);(1,3)(2,5)(4,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,21)(18,20)(19,28)(22,31)(23,29)(24,30)(25,27)(26,32)

Polycyclic presentation:

2^9|b^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8947387);