#99286

order	512
nilpotency class	3
isoclinism	isoclinic class of #99283
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#382
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

242 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eg,c^2=dg,d^2=h,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=ci,f^b=fgh,f^c=fgh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6482);