#10166327

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

4259 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)(3,8)(4,5)(6,13)(7,9)(10,12)(11,15)(14,16)(17,18)(19,27)(20,29)(21,26)(22,30)(23,25)(24,28)(31,32)(33,34,37,40)(35,41,43,38)(36,39,45,48)(42,47,44,46);(1,3)(2,8)(4,12)(5,10)(6,14)(7,11)(9,15)(13,16)(17,19,23,28)(18,24,25,27)(20,26,30,32)(21,29,31,22)(33,35)(34,38)(36,44)(37,43)(39,47)(40,41)(42,45)(46,48);(2,9)(3,11)(4,14)(10,13)(18,25)(20,30)(24,27)(26,32)(33,35,37,43)(34,38,40,41)(36,42,45,44)(39,46,48,47);(1,4,7,14)(2,5,9,16)(3,12,11,6)(8,10,15,13)(17,20)(18,22)(19,26)(21,24)(23,30)(25,29)(27,31)(28,32)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,45)(47,48);(1,5)(2,4)(3,13)(6,8)(7,16)(9,14)(10,11)(12,15)(17,21,23,31)(18,26,25,32)(19,22,28,29)(20,27,30,24)(33,36)(34,39)(35,42)(37,45)(38,46)(40,48)(41,47)(43,44);(1,6,7,12)(2,10,9,13)(3,14,11,4)(5,15,16,8)(17,22)(18,20)(19,21)(23,29)(24,26)(25,30)(27,32)(28,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=ghi,d^2=gh,e^2=hi,f^2=gh,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, 10073523);