#8837064

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	#48411	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3963 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)(3,9)(4,10)(5,13)(6,7)(8,14)(11,12)(15,16)(17,18)(19,21)(20,25)(22,30)(23,24)(26,31)(27,32)(28,29)(33,34,39,40)(35,43,44,41)(36,46,45,42)(37,38,47,48);(2,7)(4,9)(5,14)(11,16)(17,19,23,20)(18,21,24,25)(22,28,31,27)(26,32,30,29)(34,40)(35,44)(37,47)(42,46);(2,7)(3,10)(8,13)(11,16)(18,24)(19,20)(26,30)(27,28)(33,35,39,44)(34,41,40,43)(36,37,45,47)(38,46,48,42);(1,3,6,10)(2,4,7,9)(5,11,14,16)(8,12,13,15)(17,20,23,19)(18,25,24,21)(22,28,31,27)(26,32,30,29)(33,36,39,45)(34,42,40,46)(35,37,44,47)(38,43,48,41);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,21,23,25)(18,20,24,19)(22,29,31,32)(26,28,30,27)(33,37)(34,38)(35,45)(36,44)(39,47)(40,48)(41,46)(42,43);(1,5)(2,8)(3,11)(4,12)(6,14)(7,13)(9,15)(10,16)(17,22,23,31)(18,26,24,30)(19,27,20,28)(21,29,25,32)(33,38)(34,37)(35,46)(36,43)(39,48)(40,47)(41,45)(42,44)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=ghi,d^2=h,e^2=gh,f^2=gh,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, 8744260);