#8417121

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	#57023	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4096 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)(4,6)(5,8)(9,10)(11,18)(12,15)(13,20)(14,16)(17,22)(19,23)(21,24)(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);(1,2,4,6)(3,5,8,7)(9,11,15,14)(10,16,12,18)(13,19,22,21)(17,23,20,24)(27,36)(28,40)(31,35)(32,39)(38,42)(41,53)(45,47)(46,52);(1,2)(3,5)(4,6)(7,8)(9,10,15,12)(11,18,14,16)(13,17,22,20)(19,24,21,23)(25,27,30,36)(26,31,34,35)(28,37,40,50)(29,42,43,38)(32,44,39,49)(33,47,48,45)(41,54,53,51)(46,56,52,55);(1,3)(2,5)(4,8)(6,7)(9,12,15,10)(11,16,14,18)(13,20,22,17)(19,23,21,24)(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)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,28,30,40)(26,32,34,39)(27,37,36,50)(29,41,43,53)(31,44,35,49)(33,46,48,52)(38,51,42,54)(45,55,47,56);(9,14,15,11)(10,16,12,18)(13,21,22,19)(17,23,20,24)(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=gi,b^2=g,c^2=h,d^2=ghi,e^2=gi,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, 8324317);