#8596406

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

4096 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=h,e^2=gh,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, 8503602);