#10219844

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	#54498	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4057 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=g,e^2=g,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, 10127040);