#6794679

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1208 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=i,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6701875);