#8712446

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4169 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,d^2=g,e^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, 8619642);