#8112477

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

5144 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)(3,5)(4,7)(6,8)(10,13)(14,16)(17,18)(19,24)(20,21)(22,30)(23,25)(26,31)(27,29)(28,32);(2,3)(6,7)(11,15)(14,16)(17,19)(18,24)(20,23)(21,25)(22,27)(26,32)(28,31)(29,30);(1,2,5,3)(4,6,8,7)(9,10)(11,14)(12,13)(15,16)(18,25)(19,20)(26,30)(27,28);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,20,23,19)(18,21,25,24)(22,28,31,27)(26,29,30,32);(1,4,5,8)(2,6,3,7)(9,10)(11,14)(12,13)(15,16)(17,21)(18,19)(20,25)(22,29)(23,24)(26,27)(28,30)(31,32);(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,32)(25,30)

Polycyclic presentation:

2^9|c^2=g,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8019673);