#8119645

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

4062 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=h,e^2=i,f^2=g,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, 8026841);