#8934617

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
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5338 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=h,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, 8841813);