#9625454

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	#40499	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4115 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)(11,18)(12,17)(13,14)(15,19)(16,23)(20,22)(21,24)(25,26,32,35)(27,29,28,38)(30,33,44,49)(31,45,46,34)(36,41,39,47)(37,51,40,42)(43,54,55,48)(50,56,52,53);(1,2)(3,4)(5,6)(7,8)(9,11)(10,16)(12,20)(13,21)(14,22)(15,23)(17,24)(18,19)(25,27,32,28)(26,29,35,38)(30,39,44,36)(31,37,46,40)(33,47,49,41)(34,42,45,51)(43,52,55,50)(48,56,54,53);(2,6)(3,7)(9,12,15,13)(10,17,19,14)(11,20,23,21)(16,24,18,22)(25,28,32,27)(26,29,35,38)(30,39,44,36)(31,40,46,37)(33,41,49,47)(34,42,45,51)(43,52,55,50)(48,53,54,56);(3,7)(4,8)(9,13,15,12)(10,17,19,14)(11,21,23,20)(16,24,18,22)(25,29)(26,28)(27,35)(30,41)(31,42)(32,38)(33,39)(34,40)(36,49)(37,45)(43,53)(44,47)(46,51)(48,52)(50,54)(55,56);(1,3)(2,4)(5,7)(6,8)(9,11)(10,18)(12,21)(13,20)(14,22)(15,23)(16,19)(17,24)(25,30,32,44)(26,33,35,49)(27,36,28,39)(29,41,38,47)(31,43,46,55)(34,48,45,54)(37,50,40,52)(42,53,51,56);(1,4)(2,3)(5,8)(6,7)(9,14,15,17)(10,13,19,12)(11,22,23,24)(16,21,18,20)(25,31)(26,34)(27,37)(28,40)(29,42)(30,43)(32,46)(33,48)(35,45)(36,50)(38,51)(39,52)(41,53)(44,55)(47,56)(49,54)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=g,e^2=h,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, 9532650);