#10142166

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4056 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=ghi,d^2=hi,e^2=h,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, 10049362);