#8291650

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

4078 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

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, 8198846);