#8391342

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

4729 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8298538);