#9923916

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

5002 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9831112);