#8942368

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

4239 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:

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

Polycyclic presentation:

2^9|b^2=ghi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=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, 8849564);