#10219521

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	#38379	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4065 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=h,e^2=gh,f^2=g,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, 10126717);