#10420565

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	#48561	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3969 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=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, 10327761);