#6937144

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1254 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=fi,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,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, 6844340);