#8199469

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

4586 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gh,e^2=gh,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, 8106665);