#8912455

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

5119 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)(5,8)(18,23)(19,26)(25,31)(27,28);(1,2,4,6)(3,5,7,8)(9,10)(11,16)(12,14)(13,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,32)(28,29)(30,31);(1,3)(2,5)(4,7)(6,8)(9,11)(10,12)(13,14)(15,16)(19,26)(21,24)(27,28)(29,32);(11,14)(12,16)(17,19,22,26)(18,24,23,21)(20,28,30,27)(25,29,31,32);(1,2)(3,5)(4,6)(7,8)(9,12,13,16)(10,14,15,11)(17,20,22,30)(18,25,23,31)(19,27,26,28)(21,29,24,32);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,14)(13,15)(17,21,22,24)(18,26,23,19)(20,29,30,32)(25,28,31,27)

Polycyclic presentation:

2^9|b^2=h,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=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, 8819651);