#8228665

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	#45590	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4170 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)(3,8)(4,10)(5,13)(6,12)(7,9)(11,16)(14,15)(34,37)(35,36)(42,45)(43,44);(1,3,7,6)(2,8,9,12)(4,11,15,13)(5,10,16,14)(17,18,22,23)(19,21,25,30)(20,27,29,24)(26,32,31,28)(33,34,39,37)(35,41,36,40)(38,42,46,45)(43,48,44,47);(2,9)(3,6)(10,14)(11,13)(18,23)(21,30)(24,27)(28,32)(33,35,39,36)(34,40,37,41)(38,43,46,44)(42,47,45,48);(1,4,7,15)(2,10,9,14)(3,13,6,11)(5,12,16,8)(17,19,22,25)(18,21,23,30)(20,26,29,31)(24,28,27,32)(33,36,39,35)(34,41,37,40)(38,44,46,43)(42,48,45,47);(1,5,7,16)(2,11,9,13)(3,14,6,10)(4,12,15,8)(17,20,22,29)(18,24,23,27)(19,26,25,31)(21,28,30,32)(33,37,39,34)(35,41,36,40)(38,45,46,42)(43,48,44,47);(1,6,7,3)(2,12,9,8)(4,11,15,13)(5,10,16,14)(17,21)(18,25)(19,23)(20,28)(22,30)(24,31)(26,27)(29,32)(33,38,39,46)(34,42,37,45)(35,43,36,44)(40,47,41,48)

Polycyclic presentation:

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

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, 8135861);