#9243511

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	#52537	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3847 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9150707);