#8994351

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
prime radical	#51567	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4000 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8901547);