#8583406

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

5113 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8490602);