#10247636

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	#55359	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=i,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, 10154832);