#8734246

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

5436 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)(19,23)(20,29)(21,25)(22,26)(24,28)(27,32)(30,31);(2,3)(6,7)(11,15)(13,16)(17,19,22,27)(18,23,26,32)(20,21,28,31)(24,30,29,25);(1,2,5,3)(4,6,8,7)(9,10)(11,13)(12,14)(15,16)(19,27)(20,28)(23,32)(24,29);(1,3,5,2)(4,7,8,6)(9,11,12,15)(10,13,14,16)(17,20)(18,24)(19,21)(22,28)(23,30)(25,32)(26,29)(27,31);(1,4)(2,6)(3,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,32)(28,29);(17,21)(18,25)(19,28)(20,27)(22,31)(23,24)(26,30)(29,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8641442);