#9490123

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	#53687	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3987 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=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, 9397319);