#8552125

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

4005 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,e^2=h,f^2=ghi,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, 8459321);