#10173517

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

4221 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=hi,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, 10080713);