#8416956

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

4098 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)(9,10,15,14)(11,18,16,12)(13,21,22,17)(19,20,23,24)(25,26,31,28)(27,32,38,35)(29,33,41,39)(30,34,44,40)(36,45,50,49)(37,46,43,48)(42,47,53,52)(51,55,54,56);(1,3,4,7)(2,5,6,8)(9,11,15,16)(10,12,14,18)(13,19,22,23)(17,20,21,24)(26,28)(29,41)(32,35)(34,40)(36,50)(42,53)(46,48)(51,54);(1,2)(3,5)(4,6)(7,8)(9,12,15,18)(10,16,14,11)(13,20,22,24)(17,23,21,19)(25,27,31,38)(26,32,28,35)(29,36,41,50)(30,43,44,37)(33,45,39,49)(34,48,40,46)(42,54,53,51)(47,56,52,55);(1,3)(2,5)(4,7)(6,8)(9,11)(10,12)(13,19)(14,18)(15,16)(17,20)(21,24)(22,23)(25,28,31,26)(27,35,38,32)(29,33,41,39)(30,40,44,34)(36,45,50,49)(37,48,43,46)(42,47,53,52)(51,55,54,56);(1,2)(3,5)(4,6)(7,8)(9,13,15,22)(10,17,14,21)(11,19,16,23)(12,20,18,24)(25,29)(26,33)(27,36)(28,39)(30,42)(31,41)(32,45)(34,47)(35,49)(37,51)(38,50)(40,52)(43,54)(44,53)(46,55)(48,56);(9,14,15,10)(11,18,16,12)(13,21,22,17)(19,24,23,20)(25,30,31,44)(26,34,28,40)(27,37,38,43)(29,42,41,53)(32,46,35,48)(33,47,39,52)(36,51,50,54)(45,55,49,56)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=gi,e^2=i,f^2=g,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, 8324152);