#9912847

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	#53861	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4087 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)(4,6,8,7)(9,10,13,16)(11,14,17,21)(12,15,20,24)(18,22,19,23)(26,31)(27,35)(29,41)(33,43)(37,49)(39,42)(46,52)(55,56);(2,3)(6,7)(11,17)(14,21)(18,19)(22,23)(26,31)(32,36)(33,43)(34,44)(45,51)(46,52)(47,48)(55,56);(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,20)(23,24)(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);(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,27,30,35)(26,32,31,36)(28,37,40,49)(29,38,41,50)(33,45,43,51)(34,46,44,52)(39,53,42,54)(47,55,48,56);(1,3,5,2)(4,6,8,7)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(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);(1,4,5,8)(2,6,3,7)(9,12)(10,15)(11,18)(13,20)(14,22)(16,24)(17,19)(21,23)(25,29)(26,34)(27,38)(28,39)(30,41)(31,44)(32,46)(33,47)(35,50)(36,52)(37,53)(40,42)(43,48)(45,55)(49,54)(51,56)

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,d^2=h,e^2=gh,f^2=g,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, 9820043);