#9040102

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	C₂×#538	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 56 points:

(1,2,5,6)(3,4,7,8)(10,14)(12,19)(15,20)(18,23)(25,26,31,29)(27,35,36,32)(28,39,41,33)(30,42,44,34)(37,45,50,49)(38,46,43,48)(40,47,54,52)(51,56,53,55);(2,6)(3,7)(11,17)(15,20)(18,23)(22,24)(27,36)(32,35)(37,50)(38,43)(45,49)(46,48)(51,53)(55,56);(1,3)(2,4)(5,7)(6,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,27,31,36)(26,32,29,35)(28,37,41,50)(30,43,44,38)(33,45,39,49)(34,48,42,46)(40,53,54,51)(47,56,52,55);(1,4)(2,3)(5,8)(6,7)(25,28,31,41)(26,33,29,39)(27,37,36,50)(30,40,44,54)(32,45,35,49)(34,47,42,52)(38,51,43,53)(46,55,48,56);(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,29,31,26)(27,32,36,35)(28,33,41,39)(30,42,44,34)(37,49,50,45)(38,46,43,48)(40,47,54,52)(51,56,53,55);(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,30)(26,34)(27,38)(28,40)(29,42)(31,44)(32,46)(33,47)(35,48)(36,43)(37,51)(39,52)(41,54)(45,55)(49,56)(50,53)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

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, 8947298);