#9434677

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

4563 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9341873);