#9843763

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

4419 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9750959);