#8926574

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	#54421	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gh,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833770);