#10166611

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4042 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=i,e^2=gi,f^2=g,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, 10073807);