#8934655

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

4920 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)(9,10)(11,13)(12,15)(14,16)(17,18,24,26)(19,22,23,20)(21,25,31,32)(27,30,29,28);(3,7)(6,8)(11,16)(13,14)(17,19,24,23)(18,20,26,22)(21,29,31,27)(25,30,32,28);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,20,24,22)(18,19,26,23)(21,30,31,28)(25,29,32,27);(9,11,12,16)(10,14,15,13)(17,21,24,31)(18,25,26,32)(19,27,23,29)(20,28,22,30);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,16)(13,15)(17,22,24,20)(18,19,26,23)(21,28,31,30)(25,29,32,27);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,23,24,19)(18,20,26,22)(21,29,31,27)(25,28,32,30)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8841851);