#8934348

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

4917 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,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, 8841544);