#10412073

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

4320 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=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, 10319269);