#8898084

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

5114 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:

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

Polycyclic presentation:

2^9|c^2=gi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805280);