#8934763

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

5166 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,e^2=gi,f^2=h,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, 8841959);