#10166334

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

4197 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=ghi,d^2=h,e^2=ghi,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, 10073530);