#9020262

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

4233 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,d^2=h,f^2=i,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, 8927458);