#10045724

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

4270 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)(3,8)(4,9)(5,13)(6,7)(10,14)(11,15)(12,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32)(33,34)(35,39)(36,42)(37,40)(38,41)(43,47)(44,48)(45,46);(2,7)(3,10)(4,13)(12,15)(17,18,21,24)(19,22,25,29)(20,23,28,32)(26,30,27,31)(35,36)(39,42)(43,44)(47,48);(2,7)(3,10)(4,13)(12,15)(19,25)(22,29)(26,27)(30,31)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(1,3,6,10)(2,8,7,14)(4,11,13,16)(5,12,9,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,4,6,13)(2,9,7,5)(3,11,10,16)(8,15,14,12)(17,19,21,25)(18,22,24,29)(20,27,28,26)(23,31,32,30)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,47)(44,48);(1,5,6,9)(2,4,7,13)(3,12,10,15)(8,11,14,16)(17,20,21,28)(18,23,24,32)(19,26,25,27)(22,30,29,31)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|b^2=gh,c^2=i,d^2=hi,e^2=g,f^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, 9952920);