#10218695

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	#57848	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4096 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)(11,14)(12,15)(13,16)(17,20)(18,22)(19,23)(21,24)(25,26,31,36)(27,33,28,32)(29,34,43,41)(30,44,46,35)(37,49,39,47)(38,48,40,50)(42,52,45,51)(53,55,54,56);(1,2)(3,7)(4,6)(5,8)(10,11)(15,17)(16,18)(23,24)(25,27,31,28)(26,32,36,33)(29,39,43,37)(30,38,46,40)(34,49,41,47)(35,48,44,50)(42,54,45,53)(51,56,52,55);(1,3)(2,5)(4,8)(6,7)(25,28,31,27)(26,33,36,32)(29,39,43,37)(30,40,46,38)(34,49,41,47)(35,50,44,48)(42,54,45,53)(51,56,52,55);(2,6)(5,7)(9,11,14,10)(12,15,20,17)(13,18,22,16)(19,23,21,24)(25,26)(27,33)(28,32)(29,41)(30,35)(31,36)(34,43)(37,47)(38,50)(39,49)(40,48)(42,52)(44,46)(45,51)(53,55)(54,56);(1,2)(3,5)(4,6)(7,8)(9,12)(10,15)(11,17)(13,21)(14,20)(16,24)(18,23)(19,22)(25,29)(26,34)(27,37)(28,39)(30,45)(31,43)(32,47)(33,49)(35,52)(36,41)(38,54)(40,53)(42,46)(44,51)(48,56)(50,55);(1,3)(2,7)(4,8)(5,6)(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)(25,30,31,46)(26,35,36,44)(27,38,28,40)(29,42,43,45)(32,48,33,50)(34,51,41,52)(37,53,39,54)(47,55,49,56)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^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, 10125891);