#8934866

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	#49740	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4037 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)(4,10)(5,13)(6,7)(9,14)(11,12)(15,16)(17,18,22,23)(19,21,25,31)(20,27,28,24)(26,32,29,30)(33,34,37,38)(35,41,43,39)(36,44,45,40)(42,46,48,47);(2,7)(3,10)(9,13)(11,16)(18,23)(19,25)(20,28)(30,32)(33,34,37,38)(35,39,43,41)(36,40,45,44)(42,46,48,47);(2,7)(3,10)(5,14)(12,15)(19,25)(21,31)(26,29)(30,32)(34,38)(36,45)(39,41)(42,48);(1,3,6,10)(2,8,7,4)(5,11,14,16)(9,15,13,12)(17,19,22,25)(18,21,23,31)(20,29,28,26)(24,32,27,30)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,4,6,8)(2,3,7,10)(5,12,14,15)(9,11,13,16)(17,20,22,28)(18,24,23,27)(19,26,25,29)(21,30,31,32)(33,35,37,43)(34,39,38,41)(36,42,45,48)(40,46,44,47);(1,5,6,14)(2,9,7,13)(3,11,10,16)(4,12,8,15)(17,21)(18,25)(19,23)(20,30)(22,31)(24,29)(26,27)(28,32)(33,36,37,45)(34,40,38,44)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=g,e^2=gi,f^2=h,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, 8842062);