#8818163

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	#46790	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3910 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8725359);