#8107419

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

4992 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014615);