#8521597

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

4110 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)(3,9)(4,10)(5,8)(6,7)(11,16)(12,13)(14,15)(17,18,23,26)(19,24,27,21)(20,22,25,31)(28,29,32,30)(34,38)(39,45)(40,43)(46,47);(2,7)(3,10)(8,15)(11,13)(17,19,23,27)(18,24,26,21)(20,28,25,32)(22,29,31,30)(33,34)(35,39)(36,40)(37,38)(41,46)(42,45)(43,44)(47,48);(2,7)(4,9)(8,15)(12,16)(19,27)(21,24)(28,32)(29,30)(33,35,37,42)(34,39,38,45)(36,41,44,48)(40,46,43,47);(1,3)(2,4)(5,13)(6,10)(7,9)(8,16)(11,14)(12,15)(17,20)(18,22)(19,28)(21,30)(23,25)(24,29)(26,31)(27,32)(33,34)(35,39)(36,43)(37,38)(40,44)(41,47)(42,45)(46,48);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,21)(18,19)(20,30)(22,28)(23,24)(25,29)(26,27)(31,32)(33,35)(34,39)(36,41)(37,42)(38,45)(40,46)(43,47)(44,48);(1,5,6,14)(2,8,7,15)(3,11,10,13)(4,12,9,16)(17,22)(18,25)(19,29)(20,26)(21,28)(23,31)(24,32)(27,30)(33,36,37,44)(34,40,38,43)(35,41,42,48)(39,46,45,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8428793);