#8633988

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 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)(3,10)(4,12)(5,6)(7,8)(9,15)(11,14)(13,16)(33,34,39,37)(35,40,36,43)(38,41,46,45)(42,47,44,48);(1,3,7,6)(2,5,8,10)(4,13,15,11)(9,16,12,14)(17,18)(19,24)(20,21)(22,23)(25,29)(26,30)(27,32)(28,31)(34,37)(40,43)(41,45)(47,48);(2,8)(5,10)(9,12)(14,16)(18,23)(20,29)(24,31)(26,32)(33,35,39,36)(34,40,37,43)(38,42,46,44)(41,47,45,48);(1,4,7,15)(2,9,8,12)(3,11,6,13)(5,14,10,16)(17,19)(18,24)(20,26)(21,30)(22,28)(23,31)(25,27)(29,32)(33,36,39,35)(34,40,37,43)(38,42,46,44)(41,48,45,47);(1,5)(2,3)(4,14)(6,8)(7,10)(9,11)(12,13)(15,16)(17,20)(18,25)(19,26)(21,23)(22,29)(24,27)(28,32)(30,31)(33,37,39,34)(35,40,36,43)(38,45,46,41)(42,47,44,48);(1,6,7,3)(2,5,8,10)(4,11,15,13)(9,16,12,14)(17,21,22,25)(18,20,23,29)(19,27,28,30)(24,32,31,26)(33,38)(34,41)(35,42)(36,44)(37,45)(39,46)(40,47)(43,48)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8541184);