#10036865

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

4299 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

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, 9944061);