#9057918

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4111 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=ghi,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8965114);