#8509487

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

4786 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8416683);