#8889381

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

3941 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)(3,9)(4,10)(5,13)(6,7)(8,14)(11,12)(15,16)(17,18,22,24)(19,23,25,20)(21,29,31,26)(27,28,30,32)(33,34,39,37)(35,38,42,40)(36,44,46,41)(43,47,48,45);(2,7)(4,9)(5,14)(11,16)(17,19)(18,23)(20,24)(21,30)(22,25)(26,28)(27,31)(29,32)(34,37)(35,42)(41,44)(43,48);(2,7)(3,10)(8,13)(11,16)(18,24)(19,25)(26,29)(27,30)(33,35,39,42)(34,40,37,38)(36,43,46,48)(41,47,44,45);(1,3,6,10)(2,4,7,9)(5,11,14,16)(8,12,13,15)(17,19)(18,23)(20,24)(21,27)(22,25)(26,32)(28,29)(30,31)(33,36,39,46)(34,41,37,44)(35,43,42,48)(38,45,40,47);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,20,22,23)(18,25,24,19)(21,28,31,32)(26,30,29,27)(33,37,39,34)(35,38,42,40)(36,44,46,41)(43,45,48,47);(1,5,6,14)(2,8,7,13)(3,11,10,16)(4,12,9,15)(17,21,22,31)(18,26,24,29)(19,27,25,30)(20,28,23,32)(33,38)(34,35)(36,45)(37,42)(39,40)(41,43)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=g,e^2=i,f^2=gi,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, 8796577);