#8118626

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

4056 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8025822);