#9625457

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

4117 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,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, 9532653);