#10042484

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3900 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)(3,7,10,5)(4,12,14,9)(11,15,13,16)(17,18)(19,21)(20,26)(22,24)(23,25)(27,30)(28,31)(29,32)(35,42)(40,43)(44,47)(45,48);(1,3,6,10)(2,7,8,5)(4,13,14,11)(9,15,12,16)(17,19,23,22)(18,24,25,21)(20,28,29,27)(26,30,32,31)(33,34,38,39)(35,43,42,40)(36,37,41,46)(44,48,47,45);(2,8)(3,10)(4,14)(15,16)(18,25)(20,29)(21,24)(27,28)(34,39)(36,41)(40,43)(44,47);(1,4,6,14)(2,9,8,12)(3,11,10,13)(5,15,7,16)(17,20,23,29)(18,26,25,32)(19,27,22,28)(21,30,24,31)(33,35,38,42)(34,40,39,43)(36,44,41,47)(37,45,46,48);(1,3)(2,7)(4,11)(5,8)(6,10)(9,16)(12,15)(13,14)(17,21,23,24)(18,19,25,22)(20,30,29,31)(26,27,32,28)(33,36)(34,37)(35,44)(38,41)(39,46)(40,45)(42,47)(43,48);(1,5,6,7)(2,10,8,3)(4,15,14,16)(9,13,12,11)(17,22,23,19)(18,21,25,24)(20,28,29,27)(26,30,32,31)(33,37)(34,41)(35,45)(36,39)(38,46)(40,47)(42,48)(43,44)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=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, 9949680);