#9127668

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
prime radical	#38543	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4126 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,e^2=h,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, 9034864);