#8341865

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

5333 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)(11,16)(12,14)(13,15)(17,18)(19,24)(20,26)(21,22)(23,25)(27,29)(28,31)(30,32);(1,2,5,4)(3,6,7,8)(9,10,14,12)(11,16,13,15)(17,19,23,22)(18,24,25,21)(20,28,30,27)(26,31,32,29);(2,4)(6,8)(9,10,14,12)(11,15,13,16)(18,25)(19,22)(26,32)(27,28);(3,7)(6,8)(9,11,14,13)(10,15,12,16)(17,20)(18,26)(19,27)(21,31)(22,28)(23,30)(24,29)(25,32);(1,3,5,7)(2,6,4,8)(9,12,14,10)(11,15,13,16)(17,21)(18,19)(20,29)(22,25)(23,24)(26,28)(27,32)(30,31);(1,4,5,2)(3,8,7,6)(9,13,14,11)(10,15,12,16)(17,22,23,19)(18,21,25,24)(20,28,30,27)(26,31,32,29)

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=g,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8249061);