#8115363

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	#2615	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4116 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

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, 8022559);