#10142174

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

4279 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=ghi,d^2=ghi,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, 10049370);