#9716289

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

4679 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)(3,6)(4,7)(5,8)(10,13)(11,15)(17,18)(19,24)(20,21)(22,30)(23,25)(26,31)(27,29)(28,32);(1,3)(2,4)(5,7)(6,8)(9,10,14,13)(11,12,15,16)(17,19,23,28)(18,24,25,32)(20,29,31,22)(21,27,26,30);(9,11)(10,12)(13,16)(14,15)(18,25)(19,28)(20,31)(27,30);(9,12)(10,15)(11,13)(14,16)(17,20)(18,26)(19,29)(21,25)(22,28)(23,31)(24,30)(27,32);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,15)(14,16)(17,21,23,26)(18,20,25,31)(19,30,28,27)(22,32,29,24);(1,4,5,6)(2,7,8,3)(9,13,14,10)(11,16,15,12)(17,22,23,29)(18,27,25,30)(19,20,28,31)(21,24,26,32)

Polycyclic presentation:

2^9|b^2=h,e^2=i,f^2=gi,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, 9623485);