#8159020

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

4810 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8066216);