#10261497

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	#52773	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4059 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=hi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10168693);