#9606449

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

4922 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=ghi,d^2=i,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9513645);