#6480158

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470395
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1290 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gh,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di,e^c=ei,e^d=ei

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, 6387354);