#8720300

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

4107 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)(3,6)(4,14)(5,16)(7,11)(8,13)(9,15)(10,12)(17,18,23,29)(19,24,31,32)(20,26,30,25)(21,22,28,27)(33,34,37,40)(35,38,41,44)(36,39,43,46)(42,45,47,48);(1,3)(2,8)(4,5)(6,11)(7,13)(9,16)(10,14)(12,15)(17,19,23,31)(18,24,29,32)(20,22,30,27)(21,26,28,25)(35,41)(38,44)(42,47)(45,48);(1,4,7,15)(2,9,11,14)(3,5,13,12)(6,10,8,16)(17,20,23,30)(18,25,29,26)(19,27,31,22)(21,24,28,32)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,4)(2,9)(3,12)(5,13)(6,16)(7,15)(8,10)(11,14)(17,20)(18,26)(19,27)(21,24)(22,31)(23,30)(25,29)(28,32)(33,35,37,41)(34,38,40,44)(36,42,43,47)(39,45,46,48);(1,5,7,12)(2,10,11,16)(3,4,13,15)(6,9,8,14)(17,21)(18,27)(19,26)(20,24)(22,29)(23,28)(25,31)(30,32)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,6,7,8)(2,3,11,13)(4,10,15,16)(5,14,12,9)(17,22,23,27)(18,28,29,21)(19,30,31,20)(24,25,32,26)(33,36,37,43)(34,39,40,46)(35,42,41,47)(38,45,44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8627496);