#8689863

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

5029 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)(3,4)(5,7)(6,8)(17,18)(19,24)(20,27)(21,26)(22,23)(25,30)(28,32)(29,31);(1,3,6,5)(2,7,8,4)(9,10)(11,15)(12,14)(13,16)(17,19,23,21)(18,24,22,26)(20,29,30,28)(25,32,27,31);(9,11)(10,13)(12,16)(14,15)(17,20)(18,25)(19,28)(21,29)(22,27)(23,30)(24,31)(26,32);(2,8)(3,5)(10,14)(13,15)(17,21,23,19)(18,26,22,24)(20,29,30,28)(25,32,27,31);(1,4)(2,3)(5,8)(6,7)(9,11)(10,15)(12,16)(13,14)(17,22,23,18)(19,26,21,24)(20,27,30,25)(28,32,29,31);(1,5,6,3)(2,7,8,4)(9,10)(11,13)(12,14)(15,16)(17,20)(18,27)(19,29)(21,28)(22,25)(23,30)(24,31)(26,32)

Polycyclic presentation:

2^9|b^2=gh,d^2=h,e^2=h,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, 8597059);