#9874863

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	#58064	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3972 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,23,20)(19,26,21,24)(22,28,30,25)(27,31,29,32)(33,34,39,37)(35,43,44,40)(36,45,46,41)(38,42,48,47);(3,9)(7,13)(10,15)(14,16)(17,18,23,20)(19,26,21,24)(22,25,30,28)(27,32,29,31)(33,35,39,44)(34,40,37,43)(36,42,46,47)(38,41,48,45);(2,6)(7,13)(8,11)(14,16)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,34,39,37)(35,40,44,43)(36,41,46,45)(38,42,48,47);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20,23,18)(19,24,21,26)(22,28,30,25)(27,31,29,32)(33,36,39,46)(34,41,37,45)(35,42,44,47)(38,40,48,43);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,21,23,19)(18,24,20,26)(22,27,30,29)(25,32,28,31)(33,37,39,34)(35,40,44,43)(36,45,46,41)(38,42,48,47);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,38,39,48)(34,42,37,47)(35,41,44,45)(36,40,46,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gi,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=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, 9782059);