#8703439

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

5562 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:

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

Polycyclic presentation:

2^9|c^2=h,d^2=gh,e^2=gh,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, 8610635);