#8351626

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

4191 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=g,e^2=gh,f^2=g,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, 8258822);