#8180383

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

4122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=hi,d^2=g,e^2=gh,f^2=h,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, 8087579);