#8374214

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

4052 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=h,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8281410);