#8933134

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

4838 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)(3,4)(5,6)(7,8)(10,13)(11,15)(17,18,22,23)(19,21,25,32)(20,27,29,24)(26,30,28,31);(1,3,7,5)(2,6,8,4)(9,10,14,13)(11,16,15,12)(17,19)(18,21)(20,28)(22,25)(23,32)(24,30)(26,29)(27,31);(2,8)(4,6)(9,11)(10,12)(13,16)(14,15)(18,23)(19,25)(20,29)(30,31);(1,4,7,6)(2,3,8,5)(9,10,14,13)(11,12,15,16)(17,20,22,29)(18,24,23,27)(19,26,25,28)(21,31,32,30);(1,5,7,3)(2,4,8,6)(9,12,14,16)(10,15,13,11)(17,21)(18,19)(20,30)(22,32)(23,25)(24,28)(26,27)(29,31);(1,6,7,4)(2,3,8,5)(9,13,14,10)(11,16,15,12)(17,18)(19,21)(20,24)(22,23)(25,32)(26,31)(27,29)(28,30)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=h,e^2=g,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, 8840330);