#6891143

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

1337 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)(25,26,31,37)(27,38,41,33)(28,35,29,44)(30,36,32,47)(34,46,51,43)(39,49,45,52)(40,54,42,50)(48,55,53,56);(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)(26,33)(28,39)(29,45)(30,40)(31,41)(32,42)(34,48)(35,52)(36,50)(37,38)(43,55)(44,49)(46,56)(47,54)(51,53);(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)(26,34)(27,39)(29,31)(30,43)(32,46)(33,48)(35,47)(36,44)(37,51)(38,53)(40,55)(41,45)(42,56)(49,50)(52,54);(3,8)(5,7)(10,17)(12,18)(15,19)(20,24)(25,29,31,28)(26,35,37,44)(27,39,41,45)(30,43,32,46)(33,49,38,52)(34,36,51,47)(40,56,42,55)(48,54,53,50);(2,6)(5,7)(9,13,14,16)(10,18,17,12)(11,21,22,23)(15,20,19,24)(25,30,31,32)(26,36,37,47)(27,40,41,42)(28,43,29,46)(33,50,38,54)(34,44,51,35)(39,55,45,56)(48,49,53,52)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fi,d^2=i,e^2=f,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, 6798339);