#10218707

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	#53496	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4055 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)(11,17)(12,14)(13,19)(15,16)(18,22)(20,21)(23,24)(25,26,32,29)(27,33,28,36)(30,34,44,41)(31,42,45,35)(37,46,39,48)(38,50,40,47)(43,53,54,49)(51,56,52,55);(1,2)(3,7)(4,6)(5,8)(10,16)(12,22)(17,19)(20,23)(25,27,32,28)(26,33,29,36)(30,39,44,37)(31,38,45,40)(34,48,41,46)(35,47,42,50)(43,52,54,51)(49,56,53,55);(1,3)(2,5)(4,8)(6,7)(9,11,15,13)(10,17,16,19)(12,20,22,23)(14,21,18,24)(25,28,32,27)(26,33,29,36)(30,39,44,37)(31,40,45,38)(34,46,41,48)(35,47,42,50)(43,52,54,51)(49,55,53,56);(2,6)(5,7)(9,12)(10,18)(11,20)(13,23)(14,16)(15,22)(17,24)(19,21)(25,29,32,26)(27,36,28,33)(30,41,44,34)(31,42,45,35)(37,48,39,46)(38,50,40,47)(43,53,54,49)(51,56,52,55);(1,2)(3,5)(4,6)(7,8)(9,13,15,11)(10,19,16,17)(12,20,22,23)(14,21,18,24)(25,30)(26,34)(27,37)(28,39)(29,41)(31,43)(32,44)(33,46)(35,49)(36,48)(38,51)(40,52)(42,53)(45,54)(47,55)(50,56);(1,3)(2,7)(4,8)(5,6)(9,14,15,18)(10,12,16,22)(11,21,13,24)(17,20,19,23)(25,31)(26,35)(27,38)(28,40)(29,42)(30,43)(32,45)(33,47)(34,49)(36,50)(37,51)(39,52)(41,53)(44,54)(46,55)(48,56)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=h,e^2=g,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, 10125903);