#8932542

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4116 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:

(1,2,4,6)(3,5,7,8)(9,10,15,20)(11,19,14,16)(12,13,18,17)(21,24,23,22)(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);(1,3,4,7)(2,5,6,8)(9,11,15,14)(10,16,20,19)(12,23,18,21)(13,24,17,22)(27,36)(32,35)(37,40)(38,44)(46,48)(47,50)(51,53)(55,56);(1,2)(3,5)(4,6)(7,8)(9,12,15,18)(10,17,20,13)(11,21,14,23)(16,24,19,22)(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);(9,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,24)(22,23)(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)(2,5)(4,7)(6,8)(9,13,15,17)(10,18,20,12)(11,22,14,24)(16,23,19,21)(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)(3,5)(4,6)(7,8)(9,14,15,11)(10,19,20,16)(12,23,18,21)(13,24,17,22)(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=h,b^2=ghi,c^2=g,d^2=gi,e^2=i,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, 8839738);