#9625419

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	#2388	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9532615);