#10218593

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

4906 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10125789);