#8630928

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

4084 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=gi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8538124);