#8132475

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

4890 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)(6,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,24)(20,21)(22,29)(23,27)(25,32)(26,28)(30,31);(1,2)(3,6)(4,5)(7,8)(10,14)(11,15)(17,19,23,30)(18,24,27,31)(20,29,32,26)(21,22,25,28);(2,5)(6,8)(9,11,13,15)(10,12,14,16)(17,20,23,32)(18,25,27,21)(19,26,30,29)(22,31,28,24);(1,2)(3,6)(4,5)(7,8)(9,12)(10,15)(11,14)(13,16)(17,21,23,25)(18,20,27,32)(19,28,30,22)(24,26,31,29);(1,2)(3,6)(4,5)(7,8)(17,20)(18,25)(19,29)(21,27)(22,31)(23,32)(24,28)(26,30);(1,3,4,7)(2,6,5,8)(17,22,23,28)(18,26,27,29)(19,25,30,21)(20,31,32,24)

Polycyclic presentation:

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

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, 8039671);