#8709121

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=g,e^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, 8616317);