#6803831

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1206 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

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

(9,10,14,17)(11,18,20,15)(12,21,13,16)(19,23,22,24)(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);(1,2,4,5)(3,7,8,6)(9,11,14,20)(10,15,17,18)(12,19,13,22)(16,23,21,24)(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)(6,7)(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);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(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);(1,3,4,8)(2,6,5,7)(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(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=fg,b^2=fhi,c^2=i,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6711027);