#9015497

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

5105 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)(3,8)(4,7)(5,6)(9,10,14,13)(11,12,16,15)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,30)(31,32);(3,4)(7,8)(10,13)(11,16)(18,22)(23,25)(24,28)(30,31);(1,3)(2,7)(4,6)(5,8)(9,11,14,16)(10,15,13,12)(18,22)(19,26)(20,29)(30,31);(3,4)(7,8)(9,12)(10,11)(13,16)(14,15)(17,19,21,26)(18,23,22,25)(20,27,29,32)(24,30,28,31);(1,4,6,3)(2,7,5,8)(9,13,14,10)(11,12,16,15)(17,18)(19,25)(20,24)(21,22)(23,26)(27,31)(28,29)(30,32);(1,5,6,2)(3,7,4,8)(9,12)(10,11)(13,16)(14,15)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)

Polycyclic presentation:

2^9|a^2=ghi,c^2=ghi,d^2=hi,e^2=i,f^2=gh,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, 8922693);