#8688249

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

5547 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)(3,5)(4,7)(6,8)(9,10,12,14)(11,13,15,16)(17,18,22,26)(19,23,29,27)(20,30,31,24)(21,25,28,32);(2,3)(6,7)(11,15)(13,16)(17,19)(18,23)(20,21)(22,29)(24,32)(25,30)(26,27)(28,31);(1,2,5,3)(4,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,29)(28,32)(30,31);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,15)(14,16)(17,20,22,31)(18,24,26,30)(19,28,29,21)(23,25,27,32);(1,4)(2,6)(3,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,24)(21,32)(22,26)(25,28)(27,29)(30,31);(17,21,22,28)(18,25,26,32)(19,20,29,31)(23,30,27,24)

Polycyclic presentation:

2^9|a^2=gh,c^2=g,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, 8595445);