#9062110

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

4960 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8969306);