#8689316

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

4929 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8596512);