#9003076

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

4058 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8910272);