#9714613

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

5059 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)(4,6)(5,7)(10,13)(11,15)(17,18)(19,25)(20,24)(21,26)(22,23)(27,29)(28,30)(31,32);(1,3,5,4)(2,6,7,8)(9,10,14,13)(11,12,15,16)(17,18)(19,21)(20,28)(22,23)(24,30)(25,26)(27,29)(31,32);(9,11)(10,12)(13,16)(14,15)(18,23)(19,26)(24,28)(27,32);(9,12)(10,15)(11,13)(14,16)(17,19)(18,21)(20,27)(22,26)(23,25)(24,31)(28,29)(30,32);(1,2)(3,6)(4,8)(5,7)(9,12)(10,11)(13,15)(14,16)(17,20)(18,24)(19,27)(21,31)(22,30)(23,28)(25,29)(26,32);(1,4,5,3)(2,6,7,8)(9,13,14,10)(11,16,15,12)(17,21)(18,19)(20,29)(22,25)(23,26)(24,32)(27,28)(30,31)

Polycyclic presentation:

2^9|b^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=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, 9621809);