#8346307

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

4874 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,15,16,13)(17,18)(19,27)(20,24)(21,29)(22,25)(23,26)(28,32)(30,31);(1,2,6,4)(3,8,5,7)(9,11,12,16)(10,13,14,15)(17,19)(18,21)(20,28)(22,24)(23,29)(25,31)(26,27)(30,32);(1,2,6,4)(3,7,5,8)(9,11)(10,13)(12,16)(14,15)(17,20,23,31)(18,24,26,30)(19,28,29,25)(21,22,27,32);(1,3,6,5)(2,7,4,8)(9,11)(10,13)(12,16)(14,15)(17,21)(18,19)(20,22)(23,27)(24,28)(25,30)(26,29)(31,32);(1,4,6,2)(3,7,5,8)(9,10)(11,13)(12,14)(15,16)(17,22)(18,25)(19,24)(20,27)(21,31)(23,32)(26,28)(29,30);(1,5,6,3)(2,7,4,8)(17,18)(19,21)(20,30)(22,25)(23,26)(24,31)(27,29)(28,32)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=hi,e^2=hi,f^2=hi,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, 8253503);