#8119317

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

4176 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)(3,4)(5,12)(6,14)(7,8)(9,15)(10,16)(11,13)(21,29)(24,31)(27,28)(30,32)(34,39)(35,37)(36,44)(47,48);(1,3,7,5)(2,4,8,12)(6,10,15,13)(9,11,14,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,34,38,39)(35,40,37,42)(36,41,44,46)(43,47,45,48);(2,8)(3,5)(9,14)(10,13)(17,19,22,20)(18,23,25,26)(21,27,29,28)(24,30,31,32)(33,35,38,37)(34,40,39,42)(36,45,44,43)(41,48,46,47);(1,4)(2,3)(5,8)(6,11)(7,12)(9,10)(13,14)(15,16)(17,20,22,19)(18,23,25,26)(21,28,29,27)(24,30,31,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,5,7,3)(2,4,8,12)(6,13,15,10)(9,11,14,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,36,38,44)(34,41,39,46)(35,43,37,45)(40,47,42,48);(1,6)(2,9)(3,10)(4,11)(5,13)(7,15)(8,14)(12,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,37,38,35)(34,42,39,40)(36,45,44,43)(41,48,46,47)

Polycyclic presentation:

2^9|b^2=h,c^2=hi,d^2=ghi,e^2=h,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, 8026513);