#9160785

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

4095 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=hi,d^2=hi,e^2=h,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^d=egi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9067981);