#6891138

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1339 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)(11,19)(12,16)(13,18)(14,17)(15,22)(20,23)(21,24)(25,26,31,36)(27,32,40,50)(28,34,42,51)(29,35,30,45)(33,44,52,41)(37,53,38,47)(39,48,43,49)(46,54,56,55);(1,2)(3,7)(4,6)(5,8)(9,11,14,22)(10,15,17,19)(12,20,18,24)(13,23,16,21)(25,27,31,40)(26,32,36,50)(28,37,42,38)(29,43,30,39)(33,46,52,56)(34,53,51,47)(35,49,45,48)(41,55,44,54);(1,3)(2,5)(4,8)(6,7)(9,12,14,18)(10,16,17,13)(11,20,22,24)(15,21,19,23)(25,28)(26,33)(27,37)(29,41)(30,44)(31,42)(32,46)(34,45)(35,51)(36,52)(38,40)(39,54)(43,55)(47,49)(48,53)(50,56);(3,8)(5,7)(10,17)(12,18)(15,19)(20,24)(25,28)(26,34)(27,38)(29,44)(30,41)(31,42)(32,47)(33,45)(35,52)(36,51)(37,40)(39,54)(43,55)(46,49)(48,56)(50,53);(2,6)(5,7)(9,13,14,16)(10,18,17,12)(11,21,22,23)(15,20,19,24)(25,29,31,30)(26,35,36,45)(27,39,40,43)(28,41,42,44)(32,48,50,49)(33,51,52,34)(37,54,38,55)(46,53,56,47)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6798334);