#8417332

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	#2673	C₂²
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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,d^2=ghi,e^2=ghi,f^2=h,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, 8324528);