#8186863

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

4130 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)(17,18,23,22)(19,25,20,24)(21,29,30,26)(27,31,28,32)(33,34)(35,39)(36,41)(37,45)(38,40)(42,46)(43,44)(47,48);(2,3)(7,9)(8,10)(14,15)(17,19,23,20)(18,24,22,25)(21,27,30,28)(26,31,29,32)(33,35,38,41)(34,39,40,36)(37,43,46,48)(42,47,45,44);(5,13)(8,10)(11,16)(14,15)(18,22)(24,25)(26,29)(31,32)(34,40)(35,41)(42,45)(43,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,23,19)(18,25,22,24)(21,28,30,27)(26,32,29,31)(33,36,38,39)(34,41,40,35)(37,44,46,47)(42,48,45,43);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32)(33,37,38,46)(34,42,40,45)(35,43,41,48)(36,44,39,47);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22,23,18)(19,25,20,24)(21,29,30,26)(27,32,28,31)(33,34)(35,39)(36,41)(37,42)(38,40)(43,47)(44,48)(45,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=gi,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, 8094059);