#8720506

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	#2555	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3997 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)(5,14)(6,12)(7,9)(11,15)(13,16)(17,18,23,25)(19,27,21,24)(20,22,28,26)(29,30,31,32)(33,34,39,40)(35,41,36,38)(37,45,47,42)(43,46,44,48);(1,3)(2,8)(4,7)(5,13)(6,11)(9,10)(12,15)(14,16)(17,19,23,21)(18,24,25,27)(20,31,28,29)(22,30,26,32)(34,40)(35,36)(42,45)(43,44);(2,9)(3,4)(11,14)(12,13)(18,25)(20,28)(24,27)(29,31)(33,35,39,36)(34,38,40,41)(37,43,47,44)(42,46,45,48);(1,4,7,3)(2,10,9,8)(5,12,15,13)(6,11,16,14)(17,20)(18,26)(19,29)(21,31)(22,25)(23,28)(24,30)(27,32)(33,36,39,35)(34,41,40,38)(37,44,47,43)(42,48,45,46);(1,5,7,15)(2,11,9,14)(3,13,4,12)(6,10,16,8)(17,21,23,19)(18,27,25,24)(20,31,28,29)(22,30,26,32)(33,37,39,47)(34,42,40,45)(35,43,36,44)(38,46,41,48);(1,6,7,16)(2,12,9,13)(3,11,4,14)(5,8,15,10)(17,22,23,26)(18,28,25,20)(19,30,21,32)(24,29,27,31)(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=ghi,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8627702);