#10143710

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	#53566	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4007 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:

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

Polycyclic presentation:

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

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, 10050906);