#6914902

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1375 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=gi,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6822098);