#10218631

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	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4074 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)(25,26,30,31)(27,35,38,32)(28,33,39,43)(29,41,42,34)(36,47,49,44)(37,45,51,48)(40,53,52,46)(50,54,56,55);(1,2)(3,7)(4,6)(5,8)(9,11,15,21)(10,16,14,19)(12,13,20,23)(17,24,22,18)(26,31)(28,39)(32,35)(34,41)(36,49)(40,52)(45,48)(50,56);(1,3)(2,5)(4,8)(6,7)(9,12,15,20)(10,17,14,22)(11,13,21,23)(16,24,19,18)(25,26)(27,35)(28,33)(29,34)(30,31)(32,38)(36,47)(37,48)(39,43)(40,46)(41,42)(44,49)(45,51)(50,55)(52,53)(54,56);(2,6)(5,7)(9,11)(10,16)(12,13)(14,19)(15,21)(17,24)(18,22)(20,23)(25,27,30,38)(26,32,31,35)(28,36,39,49)(29,37,42,51)(33,44,43,47)(34,45,41,48)(40,50,52,56)(46,54,53,55);(1,2)(3,5)(4,6)(7,8)(9,13)(10,18)(11,20)(12,21)(14,24)(15,23)(16,17)(19,22)(25,28)(26,33)(27,36)(29,40)(30,39)(31,43)(32,44)(34,46)(35,47)(37,50)(38,49)(41,53)(42,52)(45,54)(48,55)(51,56);(1,3)(2,7)(4,8)(5,6)(9,14,15,10)(11,19,21,16)(12,22,20,17)(13,18,23,24)(25,29,30,42)(26,34,31,41)(27,37,38,51)(28,40,39,52)(32,45,35,48)(33,46,43,53)(36,50,49,56)(44,54,47,55)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=h,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, 10125827);