#8425360

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

4196 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=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, 8332556);