#8199673

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

5045 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8106869);