#8688312

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

4789 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,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, 8595508);