#10247177

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	#46827	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4071 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,d^2=hi,e^2=h,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, 10154373);