#9912939

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4126 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)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(26,31)(27,35)(29,41)(33,43)(37,49)(39,42)(46,52)(55,56);(2,3)(6,7)(10,15)(16,17)(18,21)(23,24)(26,31)(32,36)(33,43)(34,44)(45,51)(46,52)(47,48)(55,56);(1,3,5,2)(4,7,8,6)(9,11,14,12)(10,16,15,17)(13,19,22,20)(18,23,21,24)(25,26,30,31)(27,36,35,32)(28,33,40,43)(29,34,41,44)(37,51,49,45)(38,52,50,46)(39,47,42,48)(53,56,54,55);(1,4,5,8)(2,6,3,7)(9,12,14,11)(10,17,15,16)(13,20,22,19)(18,24,21,23)(25,27)(26,32)(28,37)(29,38)(30,35)(31,36)(33,45)(34,46)(39,53)(40,49)(41,50)(42,54)(43,51)(44,52)(47,55)(48,56);(9,10)(11,16)(12,17)(13,21)(14,15)(18,22)(19,24)(20,23)(25,28,30,40)(26,33,31,43)(27,37,35,49)(29,42,41,39)(32,45,36,51)(34,48,44,47)(38,54,50,53)(46,56,52,55);(9,13)(10,18)(11,19)(12,20)(14,22)(15,21)(16,23)(17,24)(25,29,30,41)(26,34,31,44)(27,38,35,50)(28,39,40,42)(32,46,36,52)(33,47,43,48)(37,53,49,54)(45,55,51,56)

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820135);