#8818214

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	#51914	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4132 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)(3,8,13,9)(4,10,15,5)(11,12,14,16)(17,18,22,27)(19,23,30,24)(20,25,26,21)(28,32,31,29);(2,7)(5,10)(8,9)(12,16)(17,19)(18,23)(20,31)(21,32)(22,30)(24,27)(25,29)(26,28)(33,34)(35,38)(36,44)(37,40)(39,45)(41,43)(42,47)(46,48);(1,3,6,13)(2,8,7,9)(4,14,15,11)(5,12,10,16)(17,19,22,30)(18,24,27,23)(20,31,26,28)(21,29,25,32)(33,34,37,40)(35,41,43,38)(36,39,45,44)(42,47,48,46);(1,3)(2,9)(4,11)(5,12)(6,13)(7,8)(10,16)(14,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,27)(25,32)(26,31)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,4)(2,10)(3,11)(5,7)(6,15)(8,12)(9,16)(13,14)(17,20,22,26)(18,25,27,21)(19,28,30,31)(23,32,24,29)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(1,5,6,10)(2,4,7,15)(3,12,13,16)(8,14,9,11)(17,21,22,25)(18,26,27,20)(19,29,30,32)(23,31,24,28)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,d^2=hi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8725410);