#8734847

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

4772 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gh,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8642043);