#8417156

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	#36032	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4103 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)(9,10,14,18)(11,19,22,15)(12,23,21,16)(13,17,20,24)(25,26,31,28)(27,32,38,35)(29,33,41,39)(30,34,44,40)(36,45,50,49)(37,46,43,48)(42,47,53,52)(51,55,54,56);(3,7)(6,8)(9,11,14,22)(10,15,18,19)(12,13,21,20)(16,17,23,24)(26,28)(29,41)(32,35)(34,40)(36,50)(42,53)(46,48)(51,54);(1,2)(3,6)(4,5)(7,8)(9,12,14,21)(10,16,18,23)(11,20,22,13)(15,24,19,17)(25,27,31,38)(26,32,28,35)(29,36,41,50)(30,43,44,37)(33,45,39,49)(34,48,40,46)(42,54,53,51)(47,56,52,55);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,24)(17,23)(18,19)(25,28,31,26)(27,35,38,32)(29,33,41,39)(30,40,44,34)(36,45,50,49)(37,48,43,46)(42,47,53,52)(51,55,54,56);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,24)(21,23)(25,29,31,41)(26,33,28,39)(27,36,38,50)(30,42,44,53)(32,45,35,49)(34,47,40,52)(37,51,43,54)(46,55,48,56);(9,13,14,20)(10,17,18,24)(11,21,22,12)(15,23,19,16)(25,30,31,44)(26,34,28,40)(27,37,38,43)(29,42,41,53)(32,46,35,48)(33,47,39,52)(36,51,50,54)(45,55,49,56)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324352);