#8709183

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4176 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,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, 8616379);