#8416898

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=i,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, 8324094);