#9040096

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	#56864	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4085 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8947292);