#8898865

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

4052 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8806061);