#9017688

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	#40730	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4257 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)(3,7)(4,12)(5,6)(8,13)(9,10)(11,15)(14,16)(18,23)(19,25)(24,30)(26,31)(33,34)(35,37)(36,45)(38,46)(39,41)(40,44)(42,43)(47,48);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,18)(19,20)(21,24)(22,23)(25,27)(26,28)(29,30)(31,32)(33,35,39,44)(34,40,41,37)(36,42,46,48)(38,43,45,47);(2,6)(3,10)(4,13)(14,15)(19,25)(20,27)(21,29)(24,30)(33,36,39,46)(34,38,41,45)(35,42,44,48)(37,47,40,43);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,19,22,25)(18,20,23,27)(21,26,29,31)(24,28,30,32)(33,36)(34,38)(35,42)(37,47)(39,46)(40,43)(41,45)(44,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,20)(18,19)(21,28)(22,27)(23,25)(24,26)(29,32)(30,31)(33,37,39,40)(34,35,41,44)(36,43,46,47)(38,48,45,42);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,21,22,29)(18,24,23,30)(19,26,25,31)(20,28,27,32)(33,38)(34,36)(35,43)(37,48)(39,45)(40,42)(41,46)(44,47)

Polycyclic presentation:

2^9|b^2=hi,c^2=hi,d^2=ghi,e^2=hi,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, 8924884);