#8746384

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3964 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)(3,8)(4,12)(5,10)(6,7)(9,14)(11,13)(15,16)(17,18)(19,23)(20,25)(21,26)(22,27)(24,28)(29,31)(30,32)(33,34)(35,40)(36,39)(37,45)(38,42)(41,47)(43,44)(46,48);(2,7)(3,11)(9,12)(10,16)(17,19)(18,23)(20,29)(21,30)(22,28)(24,27)(25,31)(26,32)(33,35,39,43)(34,40,36,44)(37,42,47,46)(38,41,48,45);(2,7)(3,11)(9,12)(10,16)(19,28)(23,24)(29,32)(30,31)(34,36)(35,43)(41,45)(42,46);(1,3,6,11)(2,8,7,13)(4,10,14,16)(5,9,15,12)(17,19)(18,24)(20,29)(21,31)(22,28)(23,27)(25,30)(26,32)(33,36,39,34)(35,40,43,44)(37,45,47,41)(38,42,48,46);(1,4,6,14)(2,9,7,12)(3,10,11,16)(5,8,15,13)(17,20)(18,25)(19,29)(21,27)(22,26)(23,31)(24,30)(28,32)(33,37)(34,41)(35,42)(36,45)(38,44)(39,47)(40,48)(43,46);(1,5,6,15)(2,10,7,16)(3,12,11,9)(4,13,14,8)(17,21,22,25)(18,26,27,20)(19,30,28,31)(23,32,24,29)(33,38,39,48)(34,42,36,46)(35,41,43,45)(37,40,47,44)

Polycyclic presentation:

2^9|b^2=gh,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

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, 8653580);