#8930532

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	#56853	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4055 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)(9,10,15,16)(11,19,20,17)(12,23,24,13)(14,21,22,18)(25,26,31,29)(27,35,36,32)(28,39,42,33)(30,43,45,34)(37,46,40,48)(38,47,44,50)(41,49,54,52)(51,56,53,55);(3,7)(6,8)(10,16)(11,20)(13,23)(18,21)(27,36)(32,35)(37,40)(38,44)(46,48)(47,50)(51,53)(55,56);(1,2)(3,6)(4,5)(7,8)(9,11,15,20)(10,17,16,19)(12,21,24,18)(13,22,23,14)(25,27)(26,32)(28,40)(29,35)(30,44)(31,36)(33,48)(34,50)(37,42)(38,45)(39,46)(41,51)(43,47)(49,55)(52,56)(53,54);(9,12)(10,13)(11,21)(14,19)(15,24)(16,23)(17,22)(18,20)(25,28,31,42)(26,33,29,39)(27,37,36,40)(30,41,45,54)(32,46,35,48)(34,49,43,52)(38,51,44,53)(47,55,50,56);(1,3,4,7)(2,6,5,8)(9,13)(10,12)(11,22)(14,20)(15,23)(16,24)(17,21)(18,19)(25,29,31,26)(27,35,36,32)(28,33,42,39)(30,43,45,34)(37,46,40,48)(38,50,44,47)(41,49,54,52)(51,55,53,56);(1,2)(3,6)(4,5)(7,8)(9,14,15,22)(10,18,16,21)(11,13,20,23)(12,19,24,17)(25,30)(26,34)(27,38)(28,41)(29,43)(31,45)(32,47)(33,49)(35,50)(36,44)(37,51)(39,52)(40,53)(42,54)(46,55)(48,56)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8837728);