#8199260

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

4848 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=h,e^2=h,f^2=g,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, 8106456);