#8934613

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

5666 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)(9,10,13,16)(11,12,15,14)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,31)(29,30);(1,3,5,8)(2,6,7,4)(9,11)(10,14)(12,16)(13,15)(17,19)(18,23)(20,28)(21,25)(22,27)(24,32)(26,29)(30,31);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,16)(13,14)(18,21)(20,29)(23,25)(26,28);(2,7)(3,8)(9,12,13,14)(10,15,16,11)(17,20,22,29)(18,24,21,30)(19,26,27,28)(23,31,25,32);(1,3)(2,4)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,19)(18,23)(20,28)(21,25)(22,27)(24,32)(26,29)(30,31);(1,4)(2,3)(5,6)(7,8)(9,11)(10,12)(13,15)(14,16)(17,21,22,18)(19,25,27,23)(20,30,29,24)(26,32,28,31)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,d^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, 8841809);