#8201713

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

5170 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)(3,8)(4,6)(5,7)(11,15)(13,16)(17,18,23,24)(19,21,27,30)(20,25,29,31)(22,26,28,32);(1,3,6,5)(2,7,4,8)(9,10,12,14)(11,13,15,16)(17,19,23,27)(18,21,24,30)(20,28,29,22)(25,32,31,26);(1,2,6,4)(3,7,5,8)(9,11)(10,13)(12,15)(14,16)(18,24)(19,27)(22,28)(25,31);(1,2,6,4)(3,8,5,7)(9,10)(11,13)(12,14)(15,16)(17,20)(18,25)(19,22)(21,26)(23,29)(24,31)(27,28)(30,32);(1,4,6,2)(3,7,5,8)(9,11)(10,13)(12,15)(14,16)(17,21)(18,19)(20,26)(22,25)(23,30)(24,27)(28,31)(29,32);(1,5,6,3)(2,7,4,8)(17,22)(18,26)(19,20)(21,25)(23,28)(24,32)(27,29)(30,31)

Polycyclic presentation:

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

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, 8108909);