#9020270

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

4254 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)(19,25)(24,31)(26,28)(33,34,38,39)(35,41,43,37)(36,44,45,40)(42,47,48,46);(1,3)(2,7)(4,11)(5,9)(6,10)(8,15)(12,13)(14,16)(17,18,22,23)(19,21,25,30)(20,24,27,31)(26,29,28,32)(33,35,38,43)(34,37,39,41)(36,42,45,48)(40,47,44,46);(3,10)(4,12)(5,7)(8,16)(19,25)(20,27)(21,30)(24,31)(34,39)(36,45)(37,41)(42,48);(1,3)(2,7)(4,13)(5,9)(6,10)(8,14)(11,12)(15,16)(17,19,22,25)(18,21,23,30)(20,28,27,26)(24,32,31,29)(33,36)(34,40)(35,42)(37,47)(38,45)(39,44)(41,46)(43,48);(1,4)(2,8)(3,11)(5,14)(6,12)(7,15)(9,16)(10,13)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,37)(34,35)(36,46)(38,41)(39,43)(40,48)(42,44)(45,47);(1,5,6,7)(2,3,9,10)(4,14,12,15)(8,11,16,13)(17,21)(18,25)(19,23)(20,29)(22,30)(24,28)(26,31)(27,32)(33,34)(35,37)(36,40)(38,39)(41,43)(42,47)(44,45)(46,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,d^2=gh,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, 8927466);