#8159867

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
prime radical	#52390	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8067063);