#9382277

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

3875 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9289473);