#10220378

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4076 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=h,f^2=gh,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, 10127574);