#8934215

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	#50043	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4107 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,6,10)(3,7,11,15)(4,13,14,8)(5,12,16,9)(17,18,23,25)(19,24,28,31)(20,21,30,26)(22,29,32,27)(34,39)(36,46)(40,41)(43,44);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,15)(12,14)(17,19,23,28)(18,24,25,31)(20,27,30,29)(21,22,26,32)(33,34)(35,40)(36,47)(37,43)(38,39)(41,45)(42,46)(44,48);(3,11)(4,14)(7,15)(8,13)(18,25)(19,28)(20,30)(22,32)(33,35,38,45)(34,40,39,41)(36,44,46,43)(37,47,48,42);(1,4)(2,8)(3,12)(5,7)(6,14)(9,11)(10,13)(15,16)(17,20,23,30)(18,26,25,21)(19,29,28,27)(22,31,32,24)(33,35)(34,41)(36,43)(37,42)(38,45)(39,40)(44,46)(47,48);(1,5)(2,9)(3,8)(4,15)(6,16)(7,14)(10,12)(11,13)(17,21,23,26)(18,20,25,30)(19,22,28,32)(24,27,31,29)(33,36)(34,42)(35,44)(37,40)(38,46)(39,47)(41,48)(43,45);(1,3)(2,7)(4,12)(5,8)(6,11)(9,14)(10,15)(13,16)(17,22)(18,27)(19,26)(20,31)(21,28)(23,32)(24,30)(25,29)(33,37,38,48)(34,43,39,44)(35,42,45,47)(36,40,46,41)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=h,e^2=h,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, 8841411);