#8417320

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	#43542	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4094 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)(4,8)(5,6)(11,12)(15,16)(19,20)(22,23)(26,31)(27,35)(28,38)(34,41)(37,49)(40,52)(45,47)(55,56);(2,6)(3,8)(9,10,14,18)(11,15,12,16)(13,17,21,24)(19,22,20,23)(26,31)(28,38)(32,43)(34,41)(36,39)(40,52)(46,51)(50,53);(1,3)(2,4)(5,8)(6,7)(9,11,14,12)(10,15,18,16)(13,19,21,20)(17,22,24,23)(25,26,30,31)(27,32,35,43)(28,33,38,44)(29,41,42,34)(36,45,39,47)(37,51,49,46)(40,54,52,48)(50,56,53,55);(1,3)(2,4)(5,8)(6,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,27,30,35)(26,32,31,43)(28,39,38,36)(29,37,42,49)(33,47,44,45)(34,46,41,51)(40,53,52,50)(48,56,54,55);(1,4)(2,3)(5,7)(6,8)(9,12,14,11)(10,16,18,15)(13,20,21,19)(17,23,24,22)(25,28)(26,33)(27,36)(29,40)(30,38)(31,44)(32,45)(34,48)(35,39)(37,50)(41,54)(42,52)(43,47)(46,55)(49,53)(51,56);(9,13,14,21)(10,17,18,24)(11,19,12,20)(15,22,16,23)(25,29,30,42)(26,34,31,41)(27,37,35,49)(28,40,38,52)(32,46,43,51)(33,48,44,54)(36,50,39,53)(45,55,47,56)

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,d^2=gi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8324516);