#8334346

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

4127 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)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,18,23,20)(19,22,25,26)(21,24,30,28)(27,29,31,32)(33,34)(35,43)(36,38)(37,41)(39,40)(42,46)(44,45)(47,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(34,40)(36,46)(41,43)(44,47);(2,6)(7,9)(8,12)(15,16)(19,25)(22,26)(27,31)(29,32)(33,35,39,37)(34,41,40,43)(36,44,46,47)(38,48,42,45);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,20,23,18)(19,26,25,22)(21,24,30,28)(27,29,31,32)(33,36)(34,42)(35,44)(37,47)(38,40)(39,46)(41,45)(43,48);(1,4)(2,8)(3,10)(5,14)(6,12)(7,15)(9,16)(11,13)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,37,39,35)(34,43,40,41)(36,44,46,47)(38,48,42,45);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,14)(13,16)(17,22,23,26)(18,25,20,19)(21,29,30,32)(24,31,28,27)(33,38,39,42)(34,36,40,46)(35,45,37,48)(41,47,43,44)

Polycyclic presentation:

2^9|a^2=ghi,c^2=g,d^2=i,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8241542);