#8821918

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

4212 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:

(1,2,6,5)(3,9,10,7)(4,8,12,13)(11,15,16,14)(17,18,22,24)(19,23,27,32)(20,25,26,21)(28,31,30,29);(2,5)(7,9)(8,13)(14,15)(17,19)(18,23)(20,30)(21,31)(22,27)(24,32)(25,29)(26,28)(33,34)(35,38)(36,44)(37,40)(39,45)(41,43)(42,47)(46,48);(3,10)(4,12)(7,9)(8,13)(18,24)(19,27)(20,26)(29,31)(33,34,37,40)(35,41,43,38)(36,39,45,44)(42,47,48,46);(1,3,6,10)(2,7,5,9)(4,11,12,16)(8,14,13,15)(17,19)(18,23)(20,28)(21,29)(22,27)(24,32)(25,31)(26,30)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,20,22,26)(18,25,24,21)(19,28,27,30)(23,31,32,29)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,5,6,2)(3,9,10,7)(4,13,12,8)(11,15,16,14)(17,21,22,25)(18,26,24,20)(19,29,27,31)(23,30,32,28)(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=hi,d^2=g,e^2=i,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, 8729114);