#8709187

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	#53500	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4162 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)(33,34,38,37)(35,42,44,39)(36,40,45,41)(43,47,48,46);(2,7)(3,10)(8,16)(11,13)(17,18,22,23)(19,21,25,30)(20,24,28,31)(26,29,32,27)(33,35)(34,39)(36,43)(37,42)(38,44)(40,46)(41,47)(45,48);(2,7)(3,10)(8,16)(11,13)(18,23)(19,25)(24,31)(26,32)(33,36,38,45)(34,40,37,41)(35,43,44,48)(39,46,42,47);(1,3)(2,4)(5,13)(6,10)(7,9)(8,14)(11,15)(12,16)(17,19)(18,21)(20,26)(22,25)(23,30)(24,29)(27,31)(28,32)(33,36)(34,41)(35,43)(37,40)(38,45)(39,47)(42,46)(44,48);(1,4)(2,3)(5,14)(6,9)(7,10)(8,13)(11,16)(12,15)(17,20)(18,24)(19,26)(21,29)(22,28)(23,31)(25,32)(27,30)(33,35)(34,42)(36,43)(37,39)(38,44)(40,47)(41,46)(45,48);(1,5,6,15)(2,8,7,16)(3,11,10,13)(4,12,9,14)(17,21)(18,25)(19,23)(20,27)(22,30)(24,26)(28,29)(31,32)(33,37,38,34)(35,42,44,39)(36,41,45,40)(43,47,48,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^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, 8616383);