#8934270

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	#48394	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=h,e^2=gh,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, 8841466);