#8179636

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

4150 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)(3,6)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(17,18,22,23)(19,21,25,30)(20,27,29,24)(26,32,31,28)(33,34)(35,42)(36,45)(37,46)(38,40)(39,41)(43,44)(47,48);(2,3)(7,9)(8,10)(14,15)(17,18,22,23)(19,21,25,30)(20,24,29,27)(26,28,31,32)(33,35,39,38)(34,40,41,42)(36,43,46,48)(37,44,45,47);(5,13)(8,10)(11,16)(14,15)(18,23)(21,30)(24,27)(28,32)(33,36)(34,37)(35,43)(38,48)(39,46)(40,44)(41,45)(42,47);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,19,22,25)(18,21,23,30)(20,26,29,31)(24,28,27,32)(33,37)(34,36)(35,44)(38,47)(39,45)(40,43)(41,46)(42,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,20,22,29)(18,24,23,27)(19,26,25,31)(21,28,30,32)(33,37)(34,36)(35,44)(38,47)(39,45)(40,43)(41,46)(42,48);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21)(18,25)(19,23)(20,28)(22,30)(24,31)(26,27)(29,32)(33,38,39,35)(34,40,41,42)(36,43,46,48)(37,47,45,44)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8086832);