#9727003

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	#48153	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4016 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,23,25)(19,28,30,24)(20,26,31,21)(22,29,32,27)(33,34,37,40)(35,41,43,38)(36,39,45,48)(42,47,44,46);(3,9)(7,13)(10,15)(14,16)(17,19,23,30)(18,24,25,28)(20,29,31,27)(21,32,26,22)(33,35)(34,38)(36,44)(37,43)(39,47)(40,41)(42,45)(46,48);(2,6)(7,13)(8,11)(14,16)(18,25)(20,31)(24,28)(27,29)(33,35,37,43)(34,38,40,41)(36,42,45,44)(39,46,48,47);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20)(18,26)(19,29)(21,25)(22,24)(23,31)(27,30)(28,32)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,45)(47,48);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,21,23,26)(18,20,25,31)(19,22,30,32)(24,27,28,29)(33,36,37,45)(34,39,40,48)(35,42,43,44)(38,46,41,47);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,22)(18,27)(19,21)(20,24)(23,32)(25,29)(26,30)(28,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9634199);