#9007902

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	#44749	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3998 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18,23,26)(19,24,21,22)(20,25,28,32)(27,31,30,29)(33,34,38,42)(35,36,40,39)(37,45,47,41)(43,48,46,44);(2,6)(7,9)(8,12)(14,15)(17,19,23,21)(18,24,26,22)(20,27,28,30)(25,31,32,29)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,48)(46,47);(2,6)(3,10)(4,13)(14,15)(19,21)(20,28)(22,24)(25,32)(33,35,38,40)(34,39,42,36)(37,46,47,43)(41,44,45,48);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,20)(18,25)(19,27)(21,30)(22,29)(23,28)(24,31)(26,32)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,48)(46,47);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,21,23,19)(18,22,26,24)(20,27,28,30)(25,31,32,29)(33,36)(34,40)(35,42)(37,44)(38,39)(41,46)(43,45)(47,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22)(18,19)(20,29)(21,26)(23,24)(25,27)(28,31)(30,32)(33,37,38,47)(34,41,42,45)(35,43,40,46)(36,44,39,48)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gh,d^2=g,e^2=ghi,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^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8915098);