#8926517

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4111 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)(9,10)(11,13)(12,20)(14,18)(15,17)(16,22)(19,23)(21,24)(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);(3,7)(5,8)(9,10,14,18)(11,13,17,15)(12,16,22,20)(19,21,24,23)(26,31)(32,43)(33,38)(34,44)(45,49)(46,48)(47,52)(55,56);(1,2)(3,5)(4,6)(7,8)(9,11,14,17)(10,15,18,13)(12,19,22,24)(16,23,20,21)(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);(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20)(25,28)(26,33)(27,35)(29,39)(30,40)(31,38)(32,45)(34,47)(36,50)(37,51)(41,53)(42,54)(43,49)(44,52)(46,55)(48,56);(1,3)(2,5)(4,7)(6,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,22)(21,24)(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);(1,2)(3,5)(4,6)(7,8)(9,13,14,15)(10,17,18,11)(12,21,22,23)(16,24,20,19)(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=g,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, 8833713);