#8158776

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3930 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=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, 8065972);