#9054435

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	#56245	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4055 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)(3,7)(4,5)(6,8)(9,10,15,12)(11,18,17,14)(13,16,22,20)(19,24,23,21)(25,26,31,34)(27,35,36,32)(28,40,43,29)(30,46,48,33)(37,38,41,44)(39,49,47,50)(42,45,55,53)(51,56,54,52);(2,5)(6,7)(9,10,15,12)(11,14,17,18)(13,20,22,16)(19,24,23,21)(27,36)(32,35)(37,41)(38,44)(39,47)(49,50)(51,54)(52,56);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,21)(20,24)(22,23)(25,27)(26,32)(28,41)(29,44)(30,47)(31,36)(33,50)(34,35)(37,43)(38,40)(39,48)(42,51)(45,52)(46,49)(53,56)(54,55);(1,3,4,8)(2,6,5,7)(9,12,15,10)(11,18,17,14)(13,20,22,16)(19,24,23,21)(25,28,31,43)(26,29,34,40)(27,37,36,41)(30,42,48,55)(32,38,35,44)(33,45,46,53)(39,51,47,54)(49,52,50,56);(9,13,15,22)(10,16,12,20)(11,19,17,23)(14,21,18,24)(25,29)(26,28)(27,38)(30,45)(31,40)(32,37)(33,42)(34,43)(35,41)(36,44)(39,52)(46,55)(47,56)(48,53)(49,51)(50,54);(9,14,15,18)(10,17,12,11)(13,21,22,24)(16,23,20,19)(25,30,31,48)(26,33,34,46)(27,39,36,47)(28,42,43,55)(29,45,40,53)(32,49,35,50)(37,51,41,54)(38,52,44,56)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

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, 8961631);