#8712469

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	#39813	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4055 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)(3,9)(4,10)(5,8)(6,7)(11,14)(12,13)(15,16)(17,18)(19,23)(20,29)(21,26)(22,24)(25,30)(27,31)(28,32)(33,34,39,43)(35,40,44,46)(36,41,45,48)(37,38,42,47);(2,7)(3,10)(8,16)(11,13)(17,19,22,27)(18,23,24,31)(20,28,30,26)(21,29,32,25)(33,35,39,44)(34,40,43,46)(36,38,45,47)(37,41,42,48);(2,7)(3,10)(8,16)(11,13)(18,24)(19,27)(25,29)(26,28)(35,44)(36,45)(40,46)(41,48);(1,3)(2,4)(5,13)(6,10)(7,9)(8,14)(11,15)(12,16)(17,19)(18,23)(20,28)(21,25)(22,27)(24,31)(26,30)(29,32)(33,36)(34,41)(35,38)(37,46)(39,45)(40,42)(43,48)(44,47);(1,4)(2,3)(5,14)(6,9)(7,10)(8,13)(11,16)(12,15)(17,20,22,30)(18,25,24,29)(19,28,27,26)(21,31,32,23)(33,37)(34,38)(35,41)(36,46)(39,42)(40,45)(43,47)(44,48);(1,5,6,15)(2,8,7,16)(3,11,10,13)(4,12,9,14)(17,21)(18,26)(19,29)(20,23)(22,32)(24,28)(25,27)(30,31)(33,38)(34,42)(35,45)(36,44)(37,43)(39,47)(40,48)(41,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,e^2=gh,f^2=g,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, 8619665);