#8425515

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

4181 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gi,e^2=h,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, 8332711);