#8552107

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

4233 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=h,e^2=gh,f^2=hi,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, 8459303);