#9002396

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

4771 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)(3,7)(4,5)(6,8)(9,10,13,11)(12,15,16,14)(17,18,22,25)(19,23,26,21)(20,24,28,31)(27,29,32,30);(3,8)(6,7)(10,11)(14,15)(17,19)(18,23)(20,27)(21,25)(22,26)(24,29)(28,32)(30,31);(1,3,5,8)(2,6,4,7)(9,11,13,10)(12,15,16,14)(19,26)(20,28)(21,23)(24,31);(3,8)(6,7)(9,12)(10,14)(11,15)(13,16)(17,20)(18,24)(19,27)(21,30)(22,28)(23,29)(25,31)(26,32);(1,3)(2,7)(4,6)(5,8)(17,21,22,23)(18,19,25,26)(20,29,28,30)(24,32,31,27);(1,4,5,2)(3,6,8,7)(17,19)(18,23)(20,27)(21,25)(22,26)(24,29)(28,32)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8909592);