#8718465

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

5025 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)(3,4,5,7)(10,13)(14,16)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,28)(31,32);(1,3,6,5)(2,7,8,4)(11,15)(14,16)(19,23)(20,26)(27,31)(28,32);(1,3,6,5)(2,7,8,4)(9,10)(11,14)(12,13)(15,16)(17,19,22,23)(18,20,25,26)(21,27,30,31)(24,28,29,32);(2,8)(3,5)(9,11)(10,14)(12,15)(13,16)(17,20,22,26)(18,23,25,19)(21,28,30,32)(24,31,29,27);(1,4)(2,3)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32);(1,5,6,3)(2,7,8,4)(17,18)(19,20)(21,29)(22,25)(23,26)(24,30)(27,32)(28,31)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8625661);