#9625463

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	#52488	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4119 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,e^2=gh,f^2=g,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, 9532659);