#8533223

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

4112 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=i,f^2=gh,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, 8440419);