#8588873

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

4174 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=h,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,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8496069);