#8864464

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

4077 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=gi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8771660);