#8436265

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

4856 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8343461);