#8688254

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

5760 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=gi,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, 8595450);