#8525477

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

5370 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,d^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, 8432673);