#9214727

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

4089 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,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, 9121923);