#7715675

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4096 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622871);