#10166421

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

4069 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=gi,e^2=gi,f^2=g,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, 10073617);