#8889409

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	#49983	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4018 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)(3,9)(4,12)(5,6)(7,10)(8,14)(11,15)(13,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32)(33,34,39,42)(35,38,45,40)(36,46,41,37)(43,44,48,47);(2,6)(7,9)(8,12)(15,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32)(33,35)(34,40)(36,43)(37,44)(38,42)(39,45)(41,48)(46,47);(3,10)(7,9)(11,13)(15,16)(18,22)(23,25)(24,28)(30,31)(33,36,39,41)(34,37,42,46)(35,43,45,48)(38,47,40,44);(1,3,5,10)(2,7,6,9)(4,13,14,11)(8,16,12,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,37,39,46)(34,41,42,36)(35,44,45,47)(38,43,40,48);(1,4)(2,8)(3,11)(5,14)(6,12)(7,15)(9,16)(10,13)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31)(33,34)(35,38)(36,37)(39,42)(40,45)(41,46)(43,47)(44,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)(33,38)(34,35)(36,47)(37,43)(39,40)(41,44)(42,45)(46,48)

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8796605);